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G = C7×C22.31C24  order 448 = 26·7

Direct product of C7 and C22.31C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×C22.31C24, C14.1122- (1+4), C14.1532+ (1+4), (C2×C28)⋊27D4, C4⋊D4⋊7C14, C4.65(D4×C14), C22⋊Q8⋊6C14, C28.472(C2×D4), C22.3(D4×C14), (C2×C14).357C24, (C2×C28).666C23, C14.192(C22×D4), C2.5(C7×2+ (1+4)), C2.4(C7×2- (1+4)), (D4×C14).216C22, C22.31(C23×C14), (C22×C14).92C23, C23.37(C22×C14), (Q8×C14).271C22, (C22×C28).447C22, (C2×C4)⋊5(C7×D4), (C14×C4⋊C4)⋊45C2, (C2×C4⋊C4)⋊18C14, C2.16(D4×C2×C14), (C2×C4○D4)⋊5C14, (C14×C4○D4)⋊21C2, (C7×C4⋊D4)⋊34C2, C4⋊C4.28(C2×C14), (C2×C14).91(C2×D4), (C7×C22⋊Q8)⋊33C2, (C2×D4).30(C2×C14), C22⋊C4.2(C2×C14), (C2×Q8).58(C2×C14), (C7×C4⋊C4).391C22, (C2×C4).24(C22×C14), (C22×C4).58(C2×C14), (C7×C22⋊C4).84C22, SmallGroup(448,1320)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.31C24
Chief series
C1 — C2 — C22 — C2×C14 — C22×C14 — D4×C14 — C7×C4⋊D4 — C7×C22.31C24
Lower central
C1 — C22 — C7×C22.31C24
Upper central
C1 — C2×C14 — C7×C22.31C24

Subgroups: 466 in 294 conjugacy classes, 162 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C7, C2×C4 [×14], C2×C4 [×10], D4 [×16], Q8 [×4], C23, C23 [×4], C14 [×3], C14 [×6], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C28 [×4], C28 [×8], C2×C14, C2×C14 [×2], C2×C14 [×14], C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8 [×4], C2×C4○D4 [×2], C2×C28 [×14], C2×C28 [×10], C7×D4 [×16], C7×Q8 [×4], C22×C14, C22×C14 [×4], C22.31C24, C7×C22⋊C4 [×8], C7×C4⋊C4 [×8], C22×C28, C22×C28 [×6], D4×C14 [×10], Q8×C14 [×2], C7×C4○D4 [×8], C14×C4⋊C4, C7×C4⋊D4 [×8], C7×C22⋊Q8 [×4], C14×C4○D4 [×2], C7×C22.31C24

Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C24, C2×C14 [×35], C22×D4, 2+ (1+4), 2- (1+4), C7×D4 [×4], C22×C14 [×15], C22.31C24, D4×C14 [×6], C23×C14, D4×C2×C14, C7×2+ (1+4), C7×2- (1+4), C7×C22.31C24

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Smallest permutation representation
►On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)

(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 221)(16 222)(17 223)(18 224)(19 218)(20 219)(21 220)(29 45)(30 46)(31 47)(32 48)(33 49)(34 43)(35 44)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 120)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 149)(142 150)(143 151)(144 152)(145 153)(146 154)(147 148)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 177)(170 178)(171 179)(172 180)(173 181)(174 182)(175 176)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 205)(198 206)(199 207)(200 208)(201 209)(202 210)(203 204)

(1 47)(2 48)(3 49)(4 43)(5 44)(6 45)(7 46)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)(162 171)(163 172)(164 173)(165 174)(166 175)(167 169)(168 170)(176 183)(177 184)(178 185)(179 186)(180 187)(181 188)(182 189)(190 199)(191 200)(192 201)(193 202)(194 203)(195 197)(196 198)(204 211)(205 212)(206 213)(207 214)(208 215)(209 216)(210 217)

(1 158)(2 159)(3 160)(4 161)(5 155)(6 156)(7 157)(8 126)(9 120)(10 121)(11 122)(12 123)(13 124)(14 125)(15 133)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 139)(37 140)(38 134)(39 135)(40 136)(41 137)(42 138)(43 154)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)

(1 47)(2 48)(3 49)(4 43)(5 44)(6 45)(7 46)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)(106 123)(107 124)(108 125)(109 126)(110 120)(111 121)(112 122)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 127)(134 151)(135 152)(136 153)(137 154)(138 148)(139 149)(140 150)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 155)(162 179)(163 180)(164 181)(165 182)(166 176)(167 177)(168 178)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 183)(190 199)(191 200)(192 201)(193 202)(194 203)(195 197)(196 198)(204 211)(205 212)(206 213)(207 214)(208 215)(209 216)(210 217)

(1 102)(2 103)(3 104)(4 105)(5 99)(6 100)(7 101)(8 189)(9 183)(10 184)(11 185)(12 186)(13 187)(14 188)(15 182)(16 176)(17 177)(18 178)(19 179)(20 180)(21 181)(22 167)(23 168)(24 162)(25 163)(26 164)(27 165)(28 166)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 83)(37 84)(38 78)(39 79)(40 80)(41 81)(42 82)(43 98)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(134 199)(135 200)(136 201)(137 202)(138 203)(139 197)(140 198)(141 195)(142 196)(143 190)(144 191)(145 192)(146 193)(147 194)(148 211)(149 212)(150 213)(151 214)(152 215)(153 216)(154 217)(155 204)(156 205)(157 206)(158 207)(159 208)(160 209)(161 210)(169 223)(170 224)(171 218)(172 219)(173 220)(174 221)(175 222)

(1 50 38 74)(2 51 39 75)(3 52 40 76)(4 53 41 77)(5 54 42 71)(6 55 36 72)(7 56 37 73)(8 202 27 210)(9 203 28 204)(10 197 22 205)(11 198 23 206)(12 199 24 207)(13 200 25 208)(14 201 26 209)(15 193 221 217)(16 194 222 211)(17 195 223 212)(18 196 224 213)(19 190 218 214)(20 191 219 215)(21 192 220 216)(29 65 45 57)(30 66 46 58)(31 67 47 59)(32 68 48 60)(33 69 49 61)(34 70 43 62)(35 64 44 63)(78 130 102 106)(79 131 103 107)(80 132 104 108)(81 133 105 109)(82 127 99 110)(83 128 100 111)(84 129 101 112)(85 121 93 113)(86 122 94 114)(87 123 95 115)(88 124 96 116)(89 125 97 117)(90 126 98 118)(91 120 92 119)(134 162 158 186)(135 163 159 187)(136 164 160 188)(137 165 161 189)(138 166 155 183)(139 167 156 184)(140 168 157 185)(141 169 149 177)(142 170 150 178)(143 171 151 179)(144 172 152 180)(145 173 153 181)(146 174 154 182)(147 175 148 176)
 
G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,176)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,205)(198,206)(199,207)(200,208)(201,209)(202,210)(203,204), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,126)(9,120)(10,121)(11,122)(12,123)(13,124)(14,125)(15,133)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,139)(37,140)(38,134)(39,135)(40,136)(41,137)(42,138)(43,154)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)(106,123)(107,124)(108,125)(109,126)(110,120)(111,121)(112,122)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,127)(134,151)(135,152)(136,153)(137,154)(138,148)(139,149)(140,150)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,179)(163,180)(164,181)(165,182)(166,176)(167,177)(168,178)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,102)(2,103)(3,104)(4,105)(5,99)(6,100)(7,101)(8,189)(9,183)(10,184)(11,185)(12,186)(13,187)(14,188)(15,182)(16,176)(17,177)(18,178)(19,179)(20,180)(21,181)(22,167)(23,168)(24,162)(25,163)(26,164)(27,165)(28,166)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,83)(37,84)(38,78)(39,79)(40,80)(41,81)(42,82)(43,98)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(134,199)(135,200)(136,201)(137,202)(138,203)(139,197)(140,198)(141,195)(142,196)(143,190)(144,191)(145,192)(146,193)(147,194)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,204)(156,205)(157,206)(158,207)(159,208)(160,209)(161,210)(169,223)(170,224)(171,218)(172,219)(173,220)(174,221)(175,222), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,202,27,210)(9,203,28,204)(10,197,22,205)(11,198,23,206)(12,199,24,207)(13,200,25,208)(14,201,26,209)(15,193,221,217)(16,194,222,211)(17,195,223,212)(18,196,224,213)(19,190,218,214)(20,191,219,215)(21,192,220,216)(29,65,45,57)(30,66,46,58)(31,67,47,59)(32,68,48,60)(33,69,49,61)(34,70,43,62)(35,64,44,63)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,121,93,113)(86,122,94,114)(87,123,95,115)(88,124,96,116)(89,125,97,117)(90,126,98,118)(91,120,92,119)(134,162,158,186)(135,163,159,187)(136,164,160,188)(137,165,161,189)(138,166,155,183)(139,167,156,184)(140,168,157,185)(141,169,149,177)(142,170,150,178)(143,171,151,179)(144,172,152,180)(145,173,153,181)(146,174,154,182)(147,175,148,176)>;
 
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,176)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,205)(198,206)(199,207)(200,208)(201,209)(202,210)(203,204), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,126)(9,120)(10,121)(11,122)(12,123)(13,124)(14,125)(15,133)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,139)(37,140)(38,134)(39,135)(40,136)(41,137)(42,138)(43,154)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)(106,123)(107,124)(108,125)(109,126)(110,120)(111,121)(112,122)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,127)(134,151)(135,152)(136,153)(137,154)(138,148)(139,149)(140,150)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,179)(163,180)(164,181)(165,182)(166,176)(167,177)(168,178)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,102)(2,103)(3,104)(4,105)(5,99)(6,100)(7,101)(8,189)(9,183)(10,184)(11,185)(12,186)(13,187)(14,188)(15,182)(16,176)(17,177)(18,178)(19,179)(20,180)(21,181)(22,167)(23,168)(24,162)(25,163)(26,164)(27,165)(28,166)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,83)(37,84)(38,78)(39,79)(40,80)(41,81)(42,82)(43,98)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(134,199)(135,200)(136,201)(137,202)(138,203)(139,197)(140,198)(141,195)(142,196)(143,190)(144,191)(145,192)(146,193)(147,194)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)(154,217)(155,204)(156,205)(157,206)(158,207)(159,208)(160,209)(161,210)(169,223)(170,224)(171,218)(172,219)(173,220)(174,221)(175,222), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,202,27,210)(9,203,28,204)(10,197,22,205)(11,198,23,206)(12,199,24,207)(13,200,25,208)(14,201,26,209)(15,193,221,217)(16,194,222,211)(17,195,223,212)(18,196,224,213)(19,190,218,214)(20,191,219,215)(21,192,220,216)(29,65,45,57)(30,66,46,58)(31,67,47,59)(32,68,48,60)(33,69,49,61)(34,70,43,62)(35,64,44,63)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,121,93,113)(86,122,94,114)(87,123,95,115)(88,124,96,116)(89,125,97,117)(90,126,98,118)(91,120,92,119)(134,162,158,186)(135,163,159,187)(136,164,160,188)(137,165,161,189)(138,166,155,183)(139,167,156,184)(140,168,157,185)(141,169,149,177)(142,170,150,178)(143,171,151,179)(144,172,152,180)(145,173,153,181)(146,174,154,182)(147,175,148,176) );
 
G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,36),(7,37),(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,221),(16,222),(17,223),(18,224),(19,218),(20,219),(21,220),(29,45),(30,46),(31,47),(32,48),(33,49),(34,43),(35,44),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,92),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,120),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,149),(142,150),(143,151),(144,152),(145,153),(146,154),(147,148),(162,186),(163,187),(164,188),(165,189),(166,183),(167,184),(168,185),(169,177),(170,178),(171,179),(172,180),(173,181),(174,182),(175,176),(190,214),(191,215),(192,216),(193,217),(194,211),(195,212),(196,213),(197,205),(198,206),(199,207),(200,208),(201,209),(202,210),(203,204)], [(1,47),(2,48),(3,49),(4,43),(5,44),(6,45),(7,46),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105),(106,115),(107,116),(108,117),(109,118),(110,119),(111,113),(112,114),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133),(134,143),(135,144),(136,145),(137,146),(138,147),(139,141),(140,142),(148,155),(149,156),(150,157),(151,158),(152,159),(153,160),(154,161),(162,171),(163,172),(164,173),(165,174),(166,175),(167,169),(168,170),(176,183),(177,184),(178,185),(179,186),(180,187),(181,188),(182,189),(190,199),(191,200),(192,201),(193,202),(194,203),(195,197),(196,198),(204,211),(205,212),(206,213),(207,214),(208,215),(209,216),(210,217)], [(1,158),(2,159),(3,160),(4,161),(5,155),(6,156),(7,157),(8,126),(9,120),(10,121),(11,122),(12,123),(13,124),(14,125),(15,133),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,139),(37,140),(38,134),(39,135),(40,136),(41,137),(42,138),(43,154),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(1,47),(2,48),(3,49),(4,43),(5,44),(6,45),(7,46),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(78,95),(79,96),(80,97),(81,98),(82,92),(83,93),(84,94),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99),(106,123),(107,124),(108,125),(109,126),(110,120),(111,121),(112,122),(113,128),(114,129),(115,130),(116,131),(117,132),(118,133),(119,127),(134,151),(135,152),(136,153),(137,154),(138,148),(139,149),(140,150),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,155),(162,179),(163,180),(164,181),(165,182),(166,176),(167,177),(168,178),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,183),(190,199),(191,200),(192,201),(193,202),(194,203),(195,197),(196,198),(204,211),(205,212),(206,213),(207,214),(208,215),(209,216),(210,217)], [(1,102),(2,103),(3,104),(4,105),(5,99),(6,100),(7,101),(8,189),(9,183),(10,184),(11,185),(12,186),(13,187),(14,188),(15,182),(16,176),(17,177),(18,178),(19,179),(20,180),(21,181),(22,167),(23,168),(24,162),(25,163),(26,164),(27,165),(28,166),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,83),(37,84),(38,78),(39,79),(40,80),(41,81),(42,82),(43,98),(44,92),(45,93),(46,94),(47,95),(48,96),(49,97),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,127),(72,128),(73,129),(74,130),(75,131),(76,132),(77,133),(134,199),(135,200),(136,201),(137,202),(138,203),(139,197),(140,198),(141,195),(142,196),(143,190),(144,191),(145,192),(146,193),(147,194),(148,211),(149,212),(150,213),(151,214),(152,215),(153,216),(154,217),(155,204),(156,205),(157,206),(158,207),(159,208),(160,209),(161,210),(169,223),(170,224),(171,218),(172,219),(173,220),(174,221),(175,222)], [(1,50,38,74),(2,51,39,75),(3,52,40,76),(4,53,41,77),(5,54,42,71),(6,55,36,72),(7,56,37,73),(8,202,27,210),(9,203,28,204),(10,197,22,205),(11,198,23,206),(12,199,24,207),(13,200,25,208),(14,201,26,209),(15,193,221,217),(16,194,222,211),(17,195,223,212),(18,196,224,213),(19,190,218,214),(20,191,219,215),(21,192,220,216),(29,65,45,57),(30,66,46,58),(31,67,47,59),(32,68,48,60),(33,69,49,61),(34,70,43,62),(35,64,44,63),(78,130,102,106),(79,131,103,107),(80,132,104,108),(81,133,105,109),(82,127,99,110),(83,128,100,111),(84,129,101,112),(85,121,93,113),(86,122,94,114),(87,123,95,115),(88,124,96,116),(89,125,97,117),(90,126,98,118),(91,120,92,119),(134,162,158,186),(135,163,159,187),(136,164,160,188),(137,165,161,189),(138,166,155,183),(139,167,156,184),(140,168,157,185),(141,169,149,177),(142,170,150,178),(143,171,151,179),(144,172,152,180),(145,173,153,181),(146,174,154,182),(147,175,148,176)])
 

Matrix representation ►G ⊆ GL6(𝔽29)

100000
010000
0023000
0002300
0000230
0000023
,
100000
010000
0028000
0002800
0000280
0000028
,
2800000
0280000
0028000
0002800
0000280
0000028
,
100000
28280000
0011111526
002424184
00271300
0020523
,
100000
010000
0028000
0002800
000010
00121201
,
120000
0280000
000010
0017172827
001000
002828012
,
2800000
0280000
000100
0028000
0017172827
0012011

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,0,0,0,0,0,0,23,0,0,0,0,0,0,23,0,0,0,0,0,0,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,28,0,0,0,0,0,28,0,0,0,0,0,0,11,24,27,2,0,0,11,24,13,0,0,0,15,18,0,5,0,0,26,4,0,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,12,0,0,0,28,0,12,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,2,28,0,0,0,0,0,0,0,17,1,28,0,0,0,17,0,28,0,0,1,28,0,0,0,0,0,27,0,12],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,17,12,0,0,1,0,17,0,0,0,0,0,28,1,0,0,0,0,27,1] >;
 

154 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4L7A···7F14A···14R14S···14AD14AE···14BB28A···28X28Y···28BT
order122222222244444···47···714···1414···1414···1428···2828···28
size111122444422224···41···11···12···24···42···24···4

154 irreducible representations

dim1111111111224444
type+++++++-
imageC1C2C2C2C2C7C14C14C14C14D4C7×D42+ (1+4)2- (1+4)C7×2+ (1+4)C7×2- (1+4)
kernelC7×C22.31C24C14×C4⋊C4C7×C4⋊D4C7×C22⋊Q8C14×C4○D4C22.31C24C2×C4⋊C4C4⋊D4C22⋊Q8C2×C4○D4C2×C28C2×C4C14C14C2C2
# reps11842664824124241166

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{31}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.31C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1320);
 
// by ID
 
G=gap.SmallGroup(448,1320);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,1227,3363,416]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=f^2=1,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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×
⋊
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