Copied to
clipboard

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×C28).666C23, (C2×C14).357C24, 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, (C22×C4).58(C2×C14), (C2×C4).24(C22×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

Generators and relations for C7×C22.31C24
 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 >

Subgroups: 466 in 294 conjugacy classes, 162 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×C4○D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C22.31C24, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C14×C4⋊C4, C7×C4⋊D4, C7×C22⋊Q8, C14×C4○D4, C7×C22.31C24
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, 2+ 1+4, 2- 1+4, C7×D4, C22×C14, C22.31C24, D4×C14, C23×C14, D4×C2×C14, C7×2+ 1+4, C7×2- 1+4, C7×C22.31C24

Smallest permutation representation of C7×C22.31C24
►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 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 127)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 155)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 183)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 211)

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

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

(1 94)(2 95)(3 96)(4 97)(5 98)(6 92)(7 93)(8 177)(9 178)(10 179)(11 180)(12 181)(13 182)(14 176)(15 175)(16 169)(17 170)(18 171)(19 172)(20 173)(21 174)(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 82)(37 83)(38 84)(39 78)(40 79)(41 80)(42 81)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(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)(134 211)(135 212)(136 213)(137 214)(138 215)(139 216)(140 217)(141 207)(142 208)(143 209)(144 210)(145 204)(146 205)(147 206)(148 201)(149 202)(150 203)(151 197)(152 198)(153 199)(154 200)(155 190)(156 191)(157 192)(158 193)(159 194)(160 195)(161 196)(183 218)(184 219)(185 220)(186 221)(187 222)(188 223)(189 224)

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

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+42- 1+4C7×2+ 1+4C7×2- 1+4
kernelC7×C22.31C24C14×C4⋊C4C7×C4⋊D4C7×C22⋊Q8C14×C4○D4C22.31C24C2×C4⋊C4C4⋊D4C22⋊Q8C2×C4○D4C2×C28C2×C4C14C14C2C2
# reps11842664824124241166

Matrix representation of C7×C22.31C24 ►in 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] >;
 

C7×C22.31C24 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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁