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G = C4⋊C4×C3×C6order 288 = 25·32

Direct product of C3×C6 and C4⋊C4

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

Aliases: C4⋊C4×C3×C6, C62.21Q8, C62.143D4, C23.16C62, C62.285C23, C42(C6×C12), (C6×C12)⋊18C4, C129(C2×C12), (C2×C12)⋊8C12, C6.82(C6×D4), C6.24(C6×Q8), (C2×C4).9C62, C62.119(C2×C4), C22.10(C6×C12), (C22×C12).34C6, C6.37(C22×C12), C22.5(C2×C62), (C6×C12).367C22, C22.3(Q8×C32), C22.13(D4×C32), (C2×C62).126C22, C2.2(D4×C3×C6), C2.1(Q8×C3×C6), C2.2(C2×C6×C12), (C2×C6×C12).7C2, (C2×C4)⋊3(C3×C12), (C3×C12)⋊26(C2×C4), (C2×C6).71(C3×D4), (C2×C6).14(C3×Q8), (C3×C6).78(C2×Q8), (C2×C12).94(C2×C6), (C2×C6).54(C2×C12), (C3×C6).299(C2×D4), (C22×C4).5(C3×C6), (C2×C6).91(C22×C6), (C22×C6).78(C2×C6), (C3×C6).129(C22×C4), SmallGroup(288,813)

Series: Derived Chief Lower central Upper central

C1C2 — C4⋊C4×C3×C6
C1C2C22C2×C6C62C6×C12C32×C4⋊C4 — C4⋊C4×C3×C6
C1C2 — C4⋊C4×C3×C6
C1C2×C62 — C4⋊C4×C3×C6

Generators and relations for C4⋊C4×C3×C6
 G = < a,b,c,d | a3=b6=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 324 in 276 conjugacy classes, 228 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×4], C4 [×4], C22, C22 [×6], C6 [×28], C2×C4 [×10], C2×C4 [×4], C23, C32, C12 [×16], C12 [×16], C2×C6 [×28], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C3×C6 [×3], C3×C6 [×4], C2×C12 [×40], C2×C12 [×16], C22×C6 [×4], C2×C4⋊C4, C3×C12 [×4], C3×C12 [×4], C62, C62 [×6], C3×C4⋊C4 [×16], C22×C12 [×12], C6×C12 [×10], C6×C12 [×4], C2×C62, C6×C4⋊C4 [×4], C32×C4⋊C4 [×4], C2×C6×C12, C2×C6×C12 [×2], C4⋊C4×C3×C6
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C32, C12 [×16], C2×C6 [×28], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C3×C6 [×7], C2×C12 [×24], C3×D4 [×8], C3×Q8 [×8], C22×C6 [×4], C2×C4⋊C4, C3×C12 [×4], C62 [×7], C3×C4⋊C4 [×16], C22×C12 [×4], C6×D4 [×4], C6×Q8 [×4], C6×C12 [×6], D4×C32 [×2], Q8×C32 [×2], C2×C62, C6×C4⋊C4 [×4], C32×C4⋊C4 [×4], C2×C6×C12, D4×C3×C6, Q8×C3×C6, C4⋊C4×C3×C6

Smallest permutation representation of C4⋊C4×C3×C6
Regular action on 288 points
Generators in S288
(1 76 80)(2 77 81)(3 78 82)(4 73 83)(5 74 84)(6 75 79)(7 288 282)(8 283 277)(9 284 278)(10 285 279)(11 286 280)(12 287 281)(13 57 31)(14 58 32)(15 59 33)(16 60 34)(17 55 35)(18 56 36)(19 27 49)(20 28 50)(21 29 51)(22 30 52)(23 25 53)(24 26 54)(37 246 45)(38 241 46)(39 242 47)(40 243 48)(41 244 43)(42 245 44)(61 111 67)(62 112 68)(63 113 69)(64 114 70)(65 109 71)(66 110 72)(85 95 103)(86 96 104)(87 91 105)(88 92 106)(89 93 107)(90 94 108)(97 147 151)(98 148 152)(99 149 153)(100 150 154)(101 145 155)(102 146 156)(115 129 123)(116 130 124)(117 131 125)(118 132 126)(119 127 121)(120 128 122)(133 183 139)(134 184 140)(135 185 141)(136 186 142)(137 181 143)(138 182 144)(157 167 175)(158 168 176)(159 163 177)(160 164 178)(161 165 179)(162 166 180)(169 219 223)(170 220 224)(171 221 225)(172 222 226)(173 217 227)(174 218 228)(187 201 195)(188 202 196)(189 203 197)(190 204 198)(191 199 193)(192 200 194)(205 255 211)(206 256 212)(207 257 213)(208 258 214)(209 253 215)(210 254 216)(229 239 247)(230 240 248)(231 235 249)(232 236 250)(233 237 251)(234 238 252)(259 273 267)(260 274 268)(261 275 269)(262 276 270)(263 271 265)(264 272 266)
(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)
(1 101 63 138)(2 102 64 133)(3 97 65 134)(4 98 66 135)(5 99 61 136)(6 100 62 137)(7 257 44 220)(8 258 45 221)(9 253 46 222)(10 254 47 217)(11 255 48 218)(12 256 43 219)(13 88 50 125)(14 89 51 126)(15 90 52 121)(16 85 53 122)(17 86 54 123)(18 87 49 124)(19 116 56 91)(20 117 57 92)(21 118 58 93)(22 119 59 94)(23 120 60 95)(24 115 55 96)(25 128 34 103)(26 129 35 104)(27 130 36 105)(28 131 31 106)(29 132 32 107)(30 127 33 108)(37 225 283 214)(38 226 284 215)(39 227 285 216)(40 228 286 211)(41 223 287 212)(42 224 288 213)(67 142 84 153)(68 143 79 154)(69 144 80 155)(70 139 81 156)(71 140 82 151)(72 141 83 152)(73 148 110 185)(74 149 111 186)(75 150 112 181)(76 145 113 182)(77 146 114 183)(78 147 109 184)(157 269 194 232)(158 270 195 233)(159 265 196 234)(160 266 197 229)(161 267 198 230)(162 268 193 231)(163 263 188 238)(164 264 189 239)(165 259 190 240)(166 260 191 235)(167 261 192 236)(168 262 187 237)(169 281 206 244)(170 282 207 245)(171 277 208 246)(172 278 209 241)(173 279 210 242)(174 280 205 243)(175 275 200 250)(176 276 201 251)(177 271 202 252)(178 272 203 247)(179 273 204 248)(180 274 199 249)
(1 210 22 166)(2 205 23 167)(3 206 24 168)(4 207 19 163)(5 208 20 164)(6 209 21 165)(7 130 252 148)(8 131 247 149)(9 132 248 150)(10 127 249 145)(11 128 250 146)(12 129 251 147)(13 197 67 225)(14 198 68 226)(15 193 69 227)(16 194 70 228)(17 195 71 223)(18 196 72 224)(25 175 77 255)(26 176 78 256)(27 177 73 257)(28 178 74 258)(29 179 75 253)(30 180 76 254)(31 203 111 221)(32 204 112 222)(33 199 113 217)(34 200 114 218)(35 201 109 219)(36 202 110 220)(37 88 266 142)(38 89 267 143)(39 90 268 144)(40 85 269 139)(41 86 270 140)(42 87 265 141)(43 104 276 184)(44 105 271 185)(45 106 272 186)(46 107 273 181)(47 108 274 182)(48 103 275 183)(49 159 83 213)(50 160 84 214)(51 161 79 215)(52 162 80 216)(53 157 81 211)(54 158 82 212)(55 187 65 169)(56 188 66 170)(57 189 61 171)(58 190 62 172)(59 191 63 173)(60 192 64 174)(91 263 135 245)(92 264 136 246)(93 259 137 241)(94 260 138 242)(95 261 133 243)(96 262 134 244)(97 281 115 237)(98 282 116 238)(99 277 117 239)(100 278 118 240)(101 279 119 235)(102 280 120 236)(121 231 155 285)(122 232 156 286)(123 233 151 287)(124 234 152 288)(125 229 153 283)(126 230 154 284)

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

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

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

180 conjugacy classes

class 1 2A···2G3A···3H4A···4L6A···6BD12A···12CR
order12···23···34···46···612···12
size11···11···12···21···12···2

180 irreducible representations

dim111111112222
type++++-
imageC1C2C2C3C4C6C6C12D4Q8C3×D4C3×Q8
kernelC4⋊C4×C3×C6C32×C4⋊C4C2×C6×C12C6×C4⋊C4C6×C12C3×C4⋊C4C22×C12C2×C12C62C62C2×C6C2×C6
# reps14388322464221616

Matrix representation of C4⋊C4×C3×C6 in GL4(𝔽13) generated by

9000
0300
0090
0009
,
3000
01200
00100
00010
,
12000
0100
0001
00120
,
8000
01200
00117
0072
G:=sub<GL(4,GF(13))| [9,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[3,0,0,0,0,12,0,0,0,0,10,0,0,0,0,10],[12,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[8,0,0,0,0,12,0,0,0,0,11,7,0,0,7,2] >;

C4⋊C4×C3×C6 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times C_3\times C_6
% in TeX

G:=Group("C4:C4xC3xC6");
// GroupNames label

G:=SmallGroup(288,813);
// by ID

G=gap.SmallGroup(288,813);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037,512]);
// Polycyclic

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

׿
×
𝔽