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G = C62.15Q8order 288 = 25·32

5th non-split extension by C62 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial

Aliases: C62.15Q8, C62.115D4, (C6×C12)⋊10C4, (C2×C12)⋊4Dic3, (C2×C6).36D12, C6.30(D6⋊C4), C62.75(C2×C4), (C3×C6).20C42, C6.16(C4×Dic3), (C2×C6).18Dic6, (C22×C12).15S3, C6.16(C4⋊Dic3), (C22×C6).145D6, C6.20(Dic3⋊C4), C32(C6.C42), C2.2(C625C4), (C2×C62).92C22, C2.2(C12⋊Dic3), C2.2(C6.11D12), C6.22(C6.D4), C326(C2.C42), C2.2(C6.Dic6), C22.11(C12⋊S3), C22.3(C324Q8), C22.16(C327D4), (C2×C6×C12).2C2, (C2×C6).50(C4×S3), (C2×C3⋊Dic3)⋊5C4, C2.5(C4×C3⋊Dic3), (C3×C6).44(C4⋊C4), (C2×C4)⋊2(C3⋊Dic3), C22.12(C4×C3⋊S3), C23.33(C2×C3⋊S3), (C2×C6).91(C3⋊D4), (C22×C4).4(C3⋊S3), (C2×C6).46(C2×Dic3), (C3×C6).61(C22⋊C4), (C22×C3⋊Dic3).2C2, C22.10(C2×C3⋊Dic3), SmallGroup(288,306)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62.15Q8
C1C3C32C3×C6C62C2×C62C22×C3⋊Dic3 — C62.15Q8
C32C3×C6 — C62.15Q8
C1C23C22×C4

Generators and relations for C62.15Q8
 G = < a,b,c,d | a6=b6=c4=1, d2=a3b3c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 668 in 228 conjugacy classes, 121 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×6], C22 [×3], C22 [×4], C6 [×28], C2×C4 [×2], C2×C4 [×10], C23, C32, Dic3 [×16], C12 [×8], C2×C6 [×28], C22×C4, C22×C4 [×2], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×32], C2×C12 [×8], C2×C12 [×8], C22×C6 [×4], C2.C42, C3⋊Dic3 [×4], C3×C12 [×2], C62 [×3], C62 [×4], C22×Dic3 [×8], C22×C12 [×4], C2×C3⋊Dic3 [×4], C2×C3⋊Dic3 [×4], C6×C12 [×2], C6×C12 [×2], C2×C62, C6.C42 [×4], C22×C3⋊Dic3 [×2], C2×C6×C12, C62.15Q8
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×4], C2×C4 [×3], D4 [×3], Q8, Dic3 [×8], D6 [×4], C42, C22⋊C4 [×3], C4⋊C4 [×3], C3⋊S3, Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×4], C3⋊D4 [×8], C2.C42, C3⋊Dic3 [×2], C2×C3⋊S3, C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×4], D6⋊C4 [×8], C6.D4 [×4], C324Q8, C4×C3⋊S3 [×2], C12⋊S3, C2×C3⋊Dic3, C327D4 [×2], C6.C42 [×4], C4×C3⋊Dic3, C6.Dic6 [×2], C12⋊Dic3, C6.11D12 [×2], C625C4, C62.15Q8

Smallest permutation representation of C62.15Q8
Regular action on 288 points
Generators in S288
(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)(223 224 225 226 227 228)(229 230 231 232 233 234)(235 236 237 238 239 240)(241 242 243 244 245 246)(247 248 249 250 251 252)(253 254 255 256 257 258)(259 260 261 262 263 264)(265 266 267 268 269 270)(271 272 273 274 275 276)(277 278 279 280 281 282)(283 284 285 286 287 288)
(1 87 67 17 28 112)(2 88 68 18 29 113)(3 89 69 13 30 114)(4 90 70 14 25 109)(5 85 71 15 26 110)(6 86 72 16 27 111)(7 219 241 288 44 263)(8 220 242 283 45 264)(9 221 243 284 46 259)(10 222 244 285 47 260)(11 217 245 286 48 261)(12 218 246 287 43 262)(19 79 38 59 62 34)(20 80 39 60 63 35)(21 81 40 55 64 36)(22 82 41 56 65 31)(23 83 42 57 66 32)(24 84 37 58 61 33)(49 280 272 240 252 268)(50 281 273 235 247 269)(51 282 274 236 248 270)(52 277 275 237 249 265)(53 278 276 238 250 266)(54 279 271 239 251 267)(73 118 135 132 92 152)(74 119 136 127 93 153)(75 120 137 128 94 154)(76 115 138 129 95 155)(77 116 133 130 96 156)(78 117 134 131 91 151)(97 185 108 157 142 126)(98 186 103 158 143 121)(99 181 104 159 144 122)(100 182 105 160 139 123)(101 183 106 161 140 124)(102 184 107 162 141 125)(145 173 214 257 189 229)(146 174 215 258 190 230)(147 169 216 253 191 231)(148 170 211 254 192 232)(149 171 212 255 187 233)(150 172 213 256 188 234)(163 175 197 226 209 201)(164 176 198 227 210 202)(165 177 193 228 205 203)(166 178 194 223 206 204)(167 179 195 224 207 199)(168 180 196 225 208 200)
(1 128 31 105)(2 129 32 106)(3 130 33 107)(4 131 34 108)(5 132 35 103)(6 127 36 104)(7 225 240 216)(8 226 235 211)(9 227 236 212)(10 228 237 213)(11 223 238 214)(12 224 239 215)(13 77 37 125)(14 78 38 126)(15 73 39 121)(16 74 40 122)(17 75 41 123)(18 76 42 124)(19 157 90 91)(20 158 85 92)(21 159 86 93)(22 160 87 94)(23 161 88 95)(24 162 89 96)(25 117 59 97)(26 118 60 98)(27 119 55 99)(28 120 56 100)(29 115 57 101)(30 116 58 102)(43 179 279 146)(44 180 280 147)(45 175 281 148)(46 176 282 149)(47 177 277 150)(48 178 278 145)(49 231 288 168)(50 232 283 163)(51 233 284 164)(52 234 285 165)(53 229 286 166)(54 230 287 167)(61 184 114 133)(62 185 109 134)(63 186 110 135)(64 181 111 136)(65 182 112 137)(66 183 113 138)(67 154 82 139)(68 155 83 140)(69 156 84 141)(70 151 79 142)(71 152 80 143)(72 153 81 144)(169 263 196 272)(170 264 197 273)(171 259 198 274)(172 260 193 275)(173 261 194 276)(174 262 195 271)(187 243 202 270)(188 244 203 265)(189 245 204 266)(190 246 199 267)(191 241 200 268)(192 242 201 269)(205 249 256 222)(206 250 257 217)(207 251 258 218)(208 252 253 219)(209 247 254 220)(210 248 255 221)
(1 198 38 190)(2 197 39 189)(3 196 40 188)(4 195 41 187)(5 194 42 192)(6 193 37 191)(7 181 52 156)(8 186 53 155)(9 185 54 154)(10 184 49 153)(11 183 50 152)(12 182 51 151)(13 200 36 172)(14 199 31 171)(15 204 32 170)(16 203 33 169)(17 202 34 174)(18 201 35 173)(19 146 28 210)(20 145 29 209)(21 150 30 208)(22 149 25 207)(23 148 26 206)(24 147 27 205)(43 160 248 117)(44 159 249 116)(45 158 250 115)(46 157 251 120)(47 162 252 119)(48 161 247 118)(55 256 89 180)(56 255 90 179)(57 254 85 178)(58 253 86 177)(59 258 87 176)(60 257 88 175)(61 216 72 165)(62 215 67 164)(63 214 68 163)(64 213 69 168)(65 212 70 167)(66 211 71 166)(73 261 106 269)(74 260 107 268)(75 259 108 267)(76 264 103 266)(77 263 104 265)(78 262 105 270)(79 230 112 227)(80 229 113 226)(81 234 114 225)(82 233 109 224)(83 232 110 223)(84 231 111 228)(91 218 100 282)(92 217 101 281)(93 222 102 280)(94 221 97 279)(95 220 98 278)(96 219 99 277)(121 276 129 242)(122 275 130 241)(123 274 131 246)(124 273 132 245)(125 272 127 244)(126 271 128 243)(133 288 144 237)(134 287 139 236)(135 286 140 235)(136 285 141 240)(137 284 142 239)(138 283 143 238)

G:=sub<Sym(288)| (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)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240)(241,242,243,244,245,246)(247,248,249,250,251,252)(253,254,255,256,257,258)(259,260,261,262,263,264)(265,266,267,268,269,270)(271,272,273,274,275,276)(277,278,279,280,281,282)(283,284,285,286,287,288), (1,87,67,17,28,112)(2,88,68,18,29,113)(3,89,69,13,30,114)(4,90,70,14,25,109)(5,85,71,15,26,110)(6,86,72,16,27,111)(7,219,241,288,44,263)(8,220,242,283,45,264)(9,221,243,284,46,259)(10,222,244,285,47,260)(11,217,245,286,48,261)(12,218,246,287,43,262)(19,79,38,59,62,34)(20,80,39,60,63,35)(21,81,40,55,64,36)(22,82,41,56,65,31)(23,83,42,57,66,32)(24,84,37,58,61,33)(49,280,272,240,252,268)(50,281,273,235,247,269)(51,282,274,236,248,270)(52,277,275,237,249,265)(53,278,276,238,250,266)(54,279,271,239,251,267)(73,118,135,132,92,152)(74,119,136,127,93,153)(75,120,137,128,94,154)(76,115,138,129,95,155)(77,116,133,130,96,156)(78,117,134,131,91,151)(97,185,108,157,142,126)(98,186,103,158,143,121)(99,181,104,159,144,122)(100,182,105,160,139,123)(101,183,106,161,140,124)(102,184,107,162,141,125)(145,173,214,257,189,229)(146,174,215,258,190,230)(147,169,216,253,191,231)(148,170,211,254,192,232)(149,171,212,255,187,233)(150,172,213,256,188,234)(163,175,197,226,209,201)(164,176,198,227,210,202)(165,177,193,228,205,203)(166,178,194,223,206,204)(167,179,195,224,207,199)(168,180,196,225,208,200), (1,128,31,105)(2,129,32,106)(3,130,33,107)(4,131,34,108)(5,132,35,103)(6,127,36,104)(7,225,240,216)(8,226,235,211)(9,227,236,212)(10,228,237,213)(11,223,238,214)(12,224,239,215)(13,77,37,125)(14,78,38,126)(15,73,39,121)(16,74,40,122)(17,75,41,123)(18,76,42,124)(19,157,90,91)(20,158,85,92)(21,159,86,93)(22,160,87,94)(23,161,88,95)(24,162,89,96)(25,117,59,97)(26,118,60,98)(27,119,55,99)(28,120,56,100)(29,115,57,101)(30,116,58,102)(43,179,279,146)(44,180,280,147)(45,175,281,148)(46,176,282,149)(47,177,277,150)(48,178,278,145)(49,231,288,168)(50,232,283,163)(51,233,284,164)(52,234,285,165)(53,229,286,166)(54,230,287,167)(61,184,114,133)(62,185,109,134)(63,186,110,135)(64,181,111,136)(65,182,112,137)(66,183,113,138)(67,154,82,139)(68,155,83,140)(69,156,84,141)(70,151,79,142)(71,152,80,143)(72,153,81,144)(169,263,196,272)(170,264,197,273)(171,259,198,274)(172,260,193,275)(173,261,194,276)(174,262,195,271)(187,243,202,270)(188,244,203,265)(189,245,204,266)(190,246,199,267)(191,241,200,268)(192,242,201,269)(205,249,256,222)(206,250,257,217)(207,251,258,218)(208,252,253,219)(209,247,254,220)(210,248,255,221), (1,198,38,190)(2,197,39,189)(3,196,40,188)(4,195,41,187)(5,194,42,192)(6,193,37,191)(7,181,52,156)(8,186,53,155)(9,185,54,154)(10,184,49,153)(11,183,50,152)(12,182,51,151)(13,200,36,172)(14,199,31,171)(15,204,32,170)(16,203,33,169)(17,202,34,174)(18,201,35,173)(19,146,28,210)(20,145,29,209)(21,150,30,208)(22,149,25,207)(23,148,26,206)(24,147,27,205)(43,160,248,117)(44,159,249,116)(45,158,250,115)(46,157,251,120)(47,162,252,119)(48,161,247,118)(55,256,89,180)(56,255,90,179)(57,254,85,178)(58,253,86,177)(59,258,87,176)(60,257,88,175)(61,216,72,165)(62,215,67,164)(63,214,68,163)(64,213,69,168)(65,212,70,167)(66,211,71,166)(73,261,106,269)(74,260,107,268)(75,259,108,267)(76,264,103,266)(77,263,104,265)(78,262,105,270)(79,230,112,227)(80,229,113,226)(81,234,114,225)(82,233,109,224)(83,232,110,223)(84,231,111,228)(91,218,100,282)(92,217,101,281)(93,222,102,280)(94,221,97,279)(95,220,98,278)(96,219,99,277)(121,276,129,242)(122,275,130,241)(123,274,131,246)(124,273,132,245)(125,272,127,244)(126,271,128,243)(133,288,144,237)(134,287,139,236)(135,286,140,235)(136,285,141,240)(137,284,142,239)(138,283,143,238)>;

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)(223,224,225,226,227,228)(229,230,231,232,233,234)(235,236,237,238,239,240)(241,242,243,244,245,246)(247,248,249,250,251,252)(253,254,255,256,257,258)(259,260,261,262,263,264)(265,266,267,268,269,270)(271,272,273,274,275,276)(277,278,279,280,281,282)(283,284,285,286,287,288), (1,87,67,17,28,112)(2,88,68,18,29,113)(3,89,69,13,30,114)(4,90,70,14,25,109)(5,85,71,15,26,110)(6,86,72,16,27,111)(7,219,241,288,44,263)(8,220,242,283,45,264)(9,221,243,284,46,259)(10,222,244,285,47,260)(11,217,245,286,48,261)(12,218,246,287,43,262)(19,79,38,59,62,34)(20,80,39,60,63,35)(21,81,40,55,64,36)(22,82,41,56,65,31)(23,83,42,57,66,32)(24,84,37,58,61,33)(49,280,272,240,252,268)(50,281,273,235,247,269)(51,282,274,236,248,270)(52,277,275,237,249,265)(53,278,276,238,250,266)(54,279,271,239,251,267)(73,118,135,132,92,152)(74,119,136,127,93,153)(75,120,137,128,94,154)(76,115,138,129,95,155)(77,116,133,130,96,156)(78,117,134,131,91,151)(97,185,108,157,142,126)(98,186,103,158,143,121)(99,181,104,159,144,122)(100,182,105,160,139,123)(101,183,106,161,140,124)(102,184,107,162,141,125)(145,173,214,257,189,229)(146,174,215,258,190,230)(147,169,216,253,191,231)(148,170,211,254,192,232)(149,171,212,255,187,233)(150,172,213,256,188,234)(163,175,197,226,209,201)(164,176,198,227,210,202)(165,177,193,228,205,203)(166,178,194,223,206,204)(167,179,195,224,207,199)(168,180,196,225,208,200), (1,128,31,105)(2,129,32,106)(3,130,33,107)(4,131,34,108)(5,132,35,103)(6,127,36,104)(7,225,240,216)(8,226,235,211)(9,227,236,212)(10,228,237,213)(11,223,238,214)(12,224,239,215)(13,77,37,125)(14,78,38,126)(15,73,39,121)(16,74,40,122)(17,75,41,123)(18,76,42,124)(19,157,90,91)(20,158,85,92)(21,159,86,93)(22,160,87,94)(23,161,88,95)(24,162,89,96)(25,117,59,97)(26,118,60,98)(27,119,55,99)(28,120,56,100)(29,115,57,101)(30,116,58,102)(43,179,279,146)(44,180,280,147)(45,175,281,148)(46,176,282,149)(47,177,277,150)(48,178,278,145)(49,231,288,168)(50,232,283,163)(51,233,284,164)(52,234,285,165)(53,229,286,166)(54,230,287,167)(61,184,114,133)(62,185,109,134)(63,186,110,135)(64,181,111,136)(65,182,112,137)(66,183,113,138)(67,154,82,139)(68,155,83,140)(69,156,84,141)(70,151,79,142)(71,152,80,143)(72,153,81,144)(169,263,196,272)(170,264,197,273)(171,259,198,274)(172,260,193,275)(173,261,194,276)(174,262,195,271)(187,243,202,270)(188,244,203,265)(189,245,204,266)(190,246,199,267)(191,241,200,268)(192,242,201,269)(205,249,256,222)(206,250,257,217)(207,251,258,218)(208,252,253,219)(209,247,254,220)(210,248,255,221), (1,198,38,190)(2,197,39,189)(3,196,40,188)(4,195,41,187)(5,194,42,192)(6,193,37,191)(7,181,52,156)(8,186,53,155)(9,185,54,154)(10,184,49,153)(11,183,50,152)(12,182,51,151)(13,200,36,172)(14,199,31,171)(15,204,32,170)(16,203,33,169)(17,202,34,174)(18,201,35,173)(19,146,28,210)(20,145,29,209)(21,150,30,208)(22,149,25,207)(23,148,26,206)(24,147,27,205)(43,160,248,117)(44,159,249,116)(45,158,250,115)(46,157,251,120)(47,162,252,119)(48,161,247,118)(55,256,89,180)(56,255,90,179)(57,254,85,178)(58,253,86,177)(59,258,87,176)(60,257,88,175)(61,216,72,165)(62,215,67,164)(63,214,68,163)(64,213,69,168)(65,212,70,167)(66,211,71,166)(73,261,106,269)(74,260,107,268)(75,259,108,267)(76,264,103,266)(77,263,104,265)(78,262,105,270)(79,230,112,227)(80,229,113,226)(81,234,114,225)(82,233,109,224)(83,232,110,223)(84,231,111,228)(91,218,100,282)(92,217,101,281)(93,222,102,280)(94,221,97,279)(95,220,98,278)(96,219,99,277)(121,276,129,242)(122,275,130,241)(123,274,131,246)(124,273,132,245)(125,272,127,244)(126,271,128,243)(133,288,144,237)(134,287,139,236)(135,286,140,235)(136,285,141,240)(137,284,142,239)(138,283,143,238) );

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),(223,224,225,226,227,228),(229,230,231,232,233,234),(235,236,237,238,239,240),(241,242,243,244,245,246),(247,248,249,250,251,252),(253,254,255,256,257,258),(259,260,261,262,263,264),(265,266,267,268,269,270),(271,272,273,274,275,276),(277,278,279,280,281,282),(283,284,285,286,287,288)], [(1,87,67,17,28,112),(2,88,68,18,29,113),(3,89,69,13,30,114),(4,90,70,14,25,109),(5,85,71,15,26,110),(6,86,72,16,27,111),(7,219,241,288,44,263),(8,220,242,283,45,264),(9,221,243,284,46,259),(10,222,244,285,47,260),(11,217,245,286,48,261),(12,218,246,287,43,262),(19,79,38,59,62,34),(20,80,39,60,63,35),(21,81,40,55,64,36),(22,82,41,56,65,31),(23,83,42,57,66,32),(24,84,37,58,61,33),(49,280,272,240,252,268),(50,281,273,235,247,269),(51,282,274,236,248,270),(52,277,275,237,249,265),(53,278,276,238,250,266),(54,279,271,239,251,267),(73,118,135,132,92,152),(74,119,136,127,93,153),(75,120,137,128,94,154),(76,115,138,129,95,155),(77,116,133,130,96,156),(78,117,134,131,91,151),(97,185,108,157,142,126),(98,186,103,158,143,121),(99,181,104,159,144,122),(100,182,105,160,139,123),(101,183,106,161,140,124),(102,184,107,162,141,125),(145,173,214,257,189,229),(146,174,215,258,190,230),(147,169,216,253,191,231),(148,170,211,254,192,232),(149,171,212,255,187,233),(150,172,213,256,188,234),(163,175,197,226,209,201),(164,176,198,227,210,202),(165,177,193,228,205,203),(166,178,194,223,206,204),(167,179,195,224,207,199),(168,180,196,225,208,200)], [(1,128,31,105),(2,129,32,106),(3,130,33,107),(4,131,34,108),(5,132,35,103),(6,127,36,104),(7,225,240,216),(8,226,235,211),(9,227,236,212),(10,228,237,213),(11,223,238,214),(12,224,239,215),(13,77,37,125),(14,78,38,126),(15,73,39,121),(16,74,40,122),(17,75,41,123),(18,76,42,124),(19,157,90,91),(20,158,85,92),(21,159,86,93),(22,160,87,94),(23,161,88,95),(24,162,89,96),(25,117,59,97),(26,118,60,98),(27,119,55,99),(28,120,56,100),(29,115,57,101),(30,116,58,102),(43,179,279,146),(44,180,280,147),(45,175,281,148),(46,176,282,149),(47,177,277,150),(48,178,278,145),(49,231,288,168),(50,232,283,163),(51,233,284,164),(52,234,285,165),(53,229,286,166),(54,230,287,167),(61,184,114,133),(62,185,109,134),(63,186,110,135),(64,181,111,136),(65,182,112,137),(66,183,113,138),(67,154,82,139),(68,155,83,140),(69,156,84,141),(70,151,79,142),(71,152,80,143),(72,153,81,144),(169,263,196,272),(170,264,197,273),(171,259,198,274),(172,260,193,275),(173,261,194,276),(174,262,195,271),(187,243,202,270),(188,244,203,265),(189,245,204,266),(190,246,199,267),(191,241,200,268),(192,242,201,269),(205,249,256,222),(206,250,257,217),(207,251,258,218),(208,252,253,219),(209,247,254,220),(210,248,255,221)], [(1,198,38,190),(2,197,39,189),(3,196,40,188),(4,195,41,187),(5,194,42,192),(6,193,37,191),(7,181,52,156),(8,186,53,155),(9,185,54,154),(10,184,49,153),(11,183,50,152),(12,182,51,151),(13,200,36,172),(14,199,31,171),(15,204,32,170),(16,203,33,169),(17,202,34,174),(18,201,35,173),(19,146,28,210),(20,145,29,209),(21,150,30,208),(22,149,25,207),(23,148,26,206),(24,147,27,205),(43,160,248,117),(44,159,249,116),(45,158,250,115),(46,157,251,120),(47,162,252,119),(48,161,247,118),(55,256,89,180),(56,255,90,179),(57,254,85,178),(58,253,86,177),(59,258,87,176),(60,257,88,175),(61,216,72,165),(62,215,67,164),(63,214,68,163),(64,213,69,168),(65,212,70,167),(66,211,71,166),(73,261,106,269),(74,260,107,268),(75,259,108,267),(76,264,103,266),(77,263,104,265),(78,262,105,270),(79,230,112,227),(80,229,113,226),(81,234,114,225),(82,233,109,224),(83,232,110,223),(84,231,111,228),(91,218,100,282),(92,217,101,281),(93,222,102,280),(94,221,97,279),(95,220,98,278),(96,219,99,277),(121,276,129,242),(122,275,130,241),(123,274,131,246),(124,273,132,245),(125,272,127,244),(126,271,128,243),(133,288,144,237),(134,287,139,236),(135,286,140,235),(136,285,141,240),(137,284,142,239),(138,283,143,238)])

84 conjugacy classes

class 1 2A···2G3A3B3C3D4A4B4C4D4E···4L6A···6AB12A···12AF
order12···2333344444···46···612···12
size11···12222222218···182···22···2

84 irreducible representations

dim11111222222222
type+++++--+-+
imageC1C2C2C4C4S3D4Q8Dic3D6Dic6C4×S3D12C3⋊D4
kernelC62.15Q8C22×C3⋊Dic3C2×C6×C12C2×C3⋊Dic3C6×C12C22×C12C62C62C2×C12C22×C6C2×C6C2×C6C2×C6C2×C6
# reps1218443184816816

Matrix representation of C62.15Q8 in GL6(𝔽13)

1200000
0120000
00121200
001000
000010
000001
,
100000
010000
0012000
0001200
000001
0000121
,
430000
390000
008000
000800
000029
0000411
,
050000
800000
0071000
003600
000029
00001111

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,2,4,0,0,0,0,9,11],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,7,3,0,0,0,0,10,6,0,0,0,0,0,0,2,11,0,0,0,0,9,11] >;

C62.15Q8 in GAP, Magma, Sage, TeX

C_6^2._{15}Q_8
% in TeX

G:=Group("C6^2.15Q8");
// GroupNames label

G:=SmallGroup(288,306);
// by ID

G=gap.SmallGroup(288,306);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,64,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=a^3*b^3*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations

׿
×
𝔽