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G = C7×C22.56C24  order 448 = 26·7

Direct product of C7 and C22.56C24

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

Aliases: C7×C22.56C24, C14.1232- (1+4), C14.1712+ (1+4), C4⋊D4⋊18C14, C22⋊Q8⋊19C14, C4.4D4⋊16C14, C42.54(C2×C14), C42.C2⋊11C14, (C2×C14).382C24, (C4×C28).295C22, (C2×C28).683C23, (D4×C14).224C22, C22.D4⋊14C14, C22.56(C23×C14), C23.25(C22×C14), (Q8×C14).187C22, C2.23(C7×2+ (1+4)), C2.15(C7×2- (1+4)), (C22×C14).108C23, (C22×C28).462C22, (C7×C4⋊D4)⋊45C2, C4⋊C4.34(C2×C14), (C7×C22⋊Q8)⋊46C2, (C2×D4).37(C2×C14), (C7×C4.4D4)⋊36C2, C22⋊C4.7(C2×C14), (C2×Q8).30(C2×C14), (C7×C42.C2)⋊28C2, (C7×C4⋊C4).251C22, (C2×C4).42(C22×C14), (C22×C4).73(C2×C14), (C7×C22.D4)⋊33C2, (C7×C22⋊C4).92C22, SmallGroup(448,1345)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.56C24
Chief series
C1 — C2 — C22 — C2×C14 — C22×C14 — D4×C14 — C7×C4.4D4 — C7×C22.56C24
Lower central
C1 — C22 — C7×C22.56C24
Upper central
C1 — C2×C14 — C7×C22.56C24

Subgroups: 362 in 220 conjugacy classes, 142 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×11], C22, C22 [×12], C7, C2×C4, C2×C4 [×10], C2×C4 [×4], D4 [×6], Q8 [×2], C23 [×4], C14, C14 [×2], C14 [×4], C42, C22⋊C4 [×12], C4⋊C4 [×10], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C28 [×11], C2×C14, C2×C14 [×12], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, C2×C28, C2×C28 [×10], C2×C28 [×4], C7×D4 [×6], C7×Q8 [×2], C22×C14 [×4], C22.56C24, C4×C28, C7×C22⋊C4 [×12], C7×C4⋊C4 [×10], C22×C28 [×4], D4×C14 [×6], Q8×C14 [×2], C7×C4⋊D4 [×4], C7×C22⋊Q8 [×4], C7×C22.D4 [×4], C7×C4.4D4 [×2], C7×C42.C2, C7×C22.56C24

Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C24, C2×C14 [×35], 2+ (1+4) [×2], 2- (1+4), C22×C14 [×15], C22.56C24, C23×C14, C7×2+ (1+4) [×2], C7×2- (1+4), C7×C22.56C24

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

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

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

Matrix representation ►G ⊆ GL8(𝔽29)

240000000
024000000
002400000
000240000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000028000
000002800
000000280
000000028
,
280000000
028000000
002800000
000280000
000028000
000002800
000000280
000000028
,
2321000000
86000000
00680000
0021230000
00000001
00000010
00000100
00001000
,
10000000
01000000
002800000
000280000
00001000
000002800
00000010
000000028
,
00100000
00010000
10000000
01000000
00000010
00000001
000028000
000002800
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(29))| [24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[23,8,0,0,0,0,0,0,21,6,0,0,0,0,0,0,0,0,6,21,0,0,0,0,0,0,8,23,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

133 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4K7A···7F14A···14R14S···14AP28A···28BN
order122222224···47···714···1414···1428···28
size111144444···41···11···14···44···4

133 irreducible representations

dim1111111111114444
type+++++++-
imageC1C2C2C2C2C2C7C14C14C14C14C142+ (1+4)2- (1+4)C7×2+ (1+4)C7×2- (1+4)
kernelC7×C22.56C24C7×C4⋊D4C7×C22⋊Q8C7×C22.D4C7×C4.4D4C7×C42.C2C22.56C24C4⋊D4C22⋊Q8C22.D4C4.4D4C42.C2C14C14C2C2
# reps144421624242412621126

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{56}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.56C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1345);
 
// by ID
 
G=gap.SmallGroup(448,1345);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,3579,604,9635,1690]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=g^2=1,f^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=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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