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## G = C12.D15order 360 = 23·32·5

### 3rd non-split extension by C12 of D15 acting via D15/C15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C30 — C12.D15
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C3⋊Dic15 — C12.D15
 Lower central C3×C15 — C3×C30 — C12.D15
 Upper central C1 — C2 — C4

Generators and relations for C12.D15
G = < a,b,c | a12=b15=1, c2=a6, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 432 in 72 conjugacy classes, 39 normal (13 characteristic)
C1, C2, C3, C4, C4, C5, C6, Q8, C32, C10, Dic3, C12, C15, C3×C6, Dic5, C20, Dic6, C30, C3⋊Dic3, C3×C12, Dic10, C3×C15, Dic15, C60, C324Q8, C3×C30, Dic30, C3⋊Dic15, C3×C60, C12.D15
Quotients: C1, C2, C22, S3, Q8, D5, D6, C3⋊S3, D10, Dic6, D15, C2×C3⋊S3, Dic10, D30, C324Q8, C3⋊D15, Dic30, C2×C3⋊D15, C12.D15

Smallest permutation representation of C12.D15
Regular action on 360 points
Generators in S360
```(1 253 22 286 304 221 68 163 209 100 195 229)(2 254 23 287 305 222 69 164 210 101 181 230)(3 255 24 288 306 223 70 165 196 102 182 231)(4 241 25 289 307 224 71 151 197 103 183 232)(5 242 26 290 308 225 72 152 198 104 184 233)(6 243 27 291 309 211 73 153 199 105 185 234)(7 244 28 292 310 212 74 154 200 91 186 235)(8 245 29 293 311 213 75 155 201 92 187 236)(9 246 30 294 312 214 61 156 202 93 188 237)(10 247 16 295 313 215 62 157 203 94 189 238)(11 248 17 296 314 216 63 158 204 95 190 239)(12 249 18 297 315 217 64 159 205 96 191 240)(13 250 19 298 301 218 65 160 206 97 192 226)(14 251 20 299 302 219 66 161 207 98 193 227)(15 252 21 300 303 220 67 162 208 99 194 228)(31 168 132 316 284 56 342 266 109 149 88 359)(32 169 133 317 285 57 343 267 110 150 89 360)(33 170 134 318 271 58 344 268 111 136 90 346)(34 171 135 319 272 59 345 269 112 137 76 347)(35 172 121 320 273 60 331 270 113 138 77 348)(36 173 122 321 274 46 332 256 114 139 78 349)(37 174 123 322 275 47 333 257 115 140 79 350)(38 175 124 323 276 48 334 258 116 141 80 351)(39 176 125 324 277 49 335 259 117 142 81 352)(40 177 126 325 278 50 336 260 118 143 82 353)(41 178 127 326 279 51 337 261 119 144 83 354)(42 179 128 327 280 52 338 262 120 145 84 355)(43 180 129 328 281 53 339 263 106 146 85 356)(44 166 130 329 282 54 340 264 107 147 86 357)(45 167 131 330 283 55 341 265 108 148 87 358)
(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)(286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312 313 314 315)(316 317 318 319 320 321 322 323 324 325 326 327 328 329 330)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345)(346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)
(1 143 68 325)(2 142 69 324)(3 141 70 323)(4 140 71 322)(5 139 72 321)(6 138 73 320)(7 137 74 319)(8 136 75 318)(9 150 61 317)(10 149 62 316)(11 148 63 330)(12 147 64 329)(13 146 65 328)(14 145 66 327)(15 144 67 326)(16 266 203 168)(17 265 204 167)(18 264 205 166)(19 263 206 180)(20 262 207 179)(21 261 208 178)(22 260 209 177)(23 259 210 176)(24 258 196 175)(25 257 197 174)(26 256 198 173)(27 270 199 172)(28 269 200 171)(29 268 201 170)(30 267 202 169)(31 295 342 94)(32 294 343 93)(33 293 344 92)(34 292 345 91)(35 291 331 105)(36 290 332 104)(37 289 333 103)(38 288 334 102)(39 287 335 101)(40 286 336 100)(41 300 337 99)(42 299 338 98)(43 298 339 97)(44 297 340 96)(45 296 341 95)(46 184 349 308)(47 183 350 307)(48 182 351 306)(49 181 352 305)(50 195 353 304)(51 194 354 303)(52 193 355 302)(53 192 356 301)(54 191 357 315)(55 190 358 314)(56 189 359 313)(57 188 360 312)(58 187 346 311)(59 186 347 310)(60 185 348 309)(76 212 272 235)(77 211 273 234)(78 225 274 233)(79 224 275 232)(80 223 276 231)(81 222 277 230)(82 221 278 229)(83 220 279 228)(84 219 280 227)(85 218 281 226)(86 217 282 240)(87 216 283 239)(88 215 284 238)(89 214 285 237)(90 213 271 236)(106 160 129 250)(107 159 130 249)(108 158 131 248)(109 157 132 247)(110 156 133 246)(111 155 134 245)(112 154 135 244)(113 153 121 243)(114 152 122 242)(115 151 123 241)(116 165 124 255)(117 164 125 254)(118 163 126 253)(119 162 127 252)(120 161 128 251)```

`G:=sub<Sym(360)| (1,253,22,286,304,221,68,163,209,100,195,229)(2,254,23,287,305,222,69,164,210,101,181,230)(3,255,24,288,306,223,70,165,196,102,182,231)(4,241,25,289,307,224,71,151,197,103,183,232)(5,242,26,290,308,225,72,152,198,104,184,233)(6,243,27,291,309,211,73,153,199,105,185,234)(7,244,28,292,310,212,74,154,200,91,186,235)(8,245,29,293,311,213,75,155,201,92,187,236)(9,246,30,294,312,214,61,156,202,93,188,237)(10,247,16,295,313,215,62,157,203,94,189,238)(11,248,17,296,314,216,63,158,204,95,190,239)(12,249,18,297,315,217,64,159,205,96,191,240)(13,250,19,298,301,218,65,160,206,97,192,226)(14,251,20,299,302,219,66,161,207,98,193,227)(15,252,21,300,303,220,67,162,208,99,194,228)(31,168,132,316,284,56,342,266,109,149,88,359)(32,169,133,317,285,57,343,267,110,150,89,360)(33,170,134,318,271,58,344,268,111,136,90,346)(34,171,135,319,272,59,345,269,112,137,76,347)(35,172,121,320,273,60,331,270,113,138,77,348)(36,173,122,321,274,46,332,256,114,139,78,349)(37,174,123,322,275,47,333,257,115,140,79,350)(38,175,124,323,276,48,334,258,116,141,80,351)(39,176,125,324,277,49,335,259,117,142,81,352)(40,177,126,325,278,50,336,260,118,143,82,353)(41,178,127,326,279,51,337,261,119,144,83,354)(42,179,128,327,280,52,338,262,120,145,84,355)(43,180,129,328,281,53,339,263,106,146,85,356)(44,166,130,329,282,54,340,264,107,147,86,357)(45,167,131,330,283,55,341,265,108,148,87,358), (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)(286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345)(346,347,348,349,350,351,352,353,354,355,356,357,358,359,360), (1,143,68,325)(2,142,69,324)(3,141,70,323)(4,140,71,322)(5,139,72,321)(6,138,73,320)(7,137,74,319)(8,136,75,318)(9,150,61,317)(10,149,62,316)(11,148,63,330)(12,147,64,329)(13,146,65,328)(14,145,66,327)(15,144,67,326)(16,266,203,168)(17,265,204,167)(18,264,205,166)(19,263,206,180)(20,262,207,179)(21,261,208,178)(22,260,209,177)(23,259,210,176)(24,258,196,175)(25,257,197,174)(26,256,198,173)(27,270,199,172)(28,269,200,171)(29,268,201,170)(30,267,202,169)(31,295,342,94)(32,294,343,93)(33,293,344,92)(34,292,345,91)(35,291,331,105)(36,290,332,104)(37,289,333,103)(38,288,334,102)(39,287,335,101)(40,286,336,100)(41,300,337,99)(42,299,338,98)(43,298,339,97)(44,297,340,96)(45,296,341,95)(46,184,349,308)(47,183,350,307)(48,182,351,306)(49,181,352,305)(50,195,353,304)(51,194,354,303)(52,193,355,302)(53,192,356,301)(54,191,357,315)(55,190,358,314)(56,189,359,313)(57,188,360,312)(58,187,346,311)(59,186,347,310)(60,185,348,309)(76,212,272,235)(77,211,273,234)(78,225,274,233)(79,224,275,232)(80,223,276,231)(81,222,277,230)(82,221,278,229)(83,220,279,228)(84,219,280,227)(85,218,281,226)(86,217,282,240)(87,216,283,239)(88,215,284,238)(89,214,285,237)(90,213,271,236)(106,160,129,250)(107,159,130,249)(108,158,131,248)(109,157,132,247)(110,156,133,246)(111,155,134,245)(112,154,135,244)(113,153,121,243)(114,152,122,242)(115,151,123,241)(116,165,124,255)(117,164,125,254)(118,163,126,253)(119,162,127,252)(120,161,128,251)>;`

`G:=Group( (1,253,22,286,304,221,68,163,209,100,195,229)(2,254,23,287,305,222,69,164,210,101,181,230)(3,255,24,288,306,223,70,165,196,102,182,231)(4,241,25,289,307,224,71,151,197,103,183,232)(5,242,26,290,308,225,72,152,198,104,184,233)(6,243,27,291,309,211,73,153,199,105,185,234)(7,244,28,292,310,212,74,154,200,91,186,235)(8,245,29,293,311,213,75,155,201,92,187,236)(9,246,30,294,312,214,61,156,202,93,188,237)(10,247,16,295,313,215,62,157,203,94,189,238)(11,248,17,296,314,216,63,158,204,95,190,239)(12,249,18,297,315,217,64,159,205,96,191,240)(13,250,19,298,301,218,65,160,206,97,192,226)(14,251,20,299,302,219,66,161,207,98,193,227)(15,252,21,300,303,220,67,162,208,99,194,228)(31,168,132,316,284,56,342,266,109,149,88,359)(32,169,133,317,285,57,343,267,110,150,89,360)(33,170,134,318,271,58,344,268,111,136,90,346)(34,171,135,319,272,59,345,269,112,137,76,347)(35,172,121,320,273,60,331,270,113,138,77,348)(36,173,122,321,274,46,332,256,114,139,78,349)(37,174,123,322,275,47,333,257,115,140,79,350)(38,175,124,323,276,48,334,258,116,141,80,351)(39,176,125,324,277,49,335,259,117,142,81,352)(40,177,126,325,278,50,336,260,118,143,82,353)(41,178,127,326,279,51,337,261,119,144,83,354)(42,179,128,327,280,52,338,262,120,145,84,355)(43,180,129,328,281,53,339,263,106,146,85,356)(44,166,130,329,282,54,340,264,107,147,86,357)(45,167,131,330,283,55,341,265,108,148,87,358), (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)(286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345)(346,347,348,349,350,351,352,353,354,355,356,357,358,359,360), (1,143,68,325)(2,142,69,324)(3,141,70,323)(4,140,71,322)(5,139,72,321)(6,138,73,320)(7,137,74,319)(8,136,75,318)(9,150,61,317)(10,149,62,316)(11,148,63,330)(12,147,64,329)(13,146,65,328)(14,145,66,327)(15,144,67,326)(16,266,203,168)(17,265,204,167)(18,264,205,166)(19,263,206,180)(20,262,207,179)(21,261,208,178)(22,260,209,177)(23,259,210,176)(24,258,196,175)(25,257,197,174)(26,256,198,173)(27,270,199,172)(28,269,200,171)(29,268,201,170)(30,267,202,169)(31,295,342,94)(32,294,343,93)(33,293,344,92)(34,292,345,91)(35,291,331,105)(36,290,332,104)(37,289,333,103)(38,288,334,102)(39,287,335,101)(40,286,336,100)(41,300,337,99)(42,299,338,98)(43,298,339,97)(44,297,340,96)(45,296,341,95)(46,184,349,308)(47,183,350,307)(48,182,351,306)(49,181,352,305)(50,195,353,304)(51,194,354,303)(52,193,355,302)(53,192,356,301)(54,191,357,315)(55,190,358,314)(56,189,359,313)(57,188,360,312)(58,187,346,311)(59,186,347,310)(60,185,348,309)(76,212,272,235)(77,211,273,234)(78,225,274,233)(79,224,275,232)(80,223,276,231)(81,222,277,230)(82,221,278,229)(83,220,279,228)(84,219,280,227)(85,218,281,226)(86,217,282,240)(87,216,283,239)(88,215,284,238)(89,214,285,237)(90,213,271,236)(106,160,129,250)(107,159,130,249)(108,158,131,248)(109,157,132,247)(110,156,133,246)(111,155,134,245)(112,154,135,244)(113,153,121,243)(114,152,122,242)(115,151,123,241)(116,165,124,255)(117,164,125,254)(118,163,126,253)(119,162,127,252)(120,161,128,251) );`

`G=PermutationGroup([[(1,253,22,286,304,221,68,163,209,100,195,229),(2,254,23,287,305,222,69,164,210,101,181,230),(3,255,24,288,306,223,70,165,196,102,182,231),(4,241,25,289,307,224,71,151,197,103,183,232),(5,242,26,290,308,225,72,152,198,104,184,233),(6,243,27,291,309,211,73,153,199,105,185,234),(7,244,28,292,310,212,74,154,200,91,186,235),(8,245,29,293,311,213,75,155,201,92,187,236),(9,246,30,294,312,214,61,156,202,93,188,237),(10,247,16,295,313,215,62,157,203,94,189,238),(11,248,17,296,314,216,63,158,204,95,190,239),(12,249,18,297,315,217,64,159,205,96,191,240),(13,250,19,298,301,218,65,160,206,97,192,226),(14,251,20,299,302,219,66,161,207,98,193,227),(15,252,21,300,303,220,67,162,208,99,194,228),(31,168,132,316,284,56,342,266,109,149,88,359),(32,169,133,317,285,57,343,267,110,150,89,360),(33,170,134,318,271,58,344,268,111,136,90,346),(34,171,135,319,272,59,345,269,112,137,76,347),(35,172,121,320,273,60,331,270,113,138,77,348),(36,173,122,321,274,46,332,256,114,139,78,349),(37,174,123,322,275,47,333,257,115,140,79,350),(38,175,124,323,276,48,334,258,116,141,80,351),(39,176,125,324,277,49,335,259,117,142,81,352),(40,177,126,325,278,50,336,260,118,143,82,353),(41,178,127,326,279,51,337,261,119,144,83,354),(42,179,128,327,280,52,338,262,120,145,84,355),(43,180,129,328,281,53,339,263,106,146,85,356),(44,166,130,329,282,54,340,264,107,147,86,357),(45,167,131,330,283,55,341,265,108,148,87,358)], [(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),(286,287,288,289,290,291,292,293,294,295,296,297,298,299,300),(301,302,303,304,305,306,307,308,309,310,311,312,313,314,315),(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330),(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345),(346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)], [(1,143,68,325),(2,142,69,324),(3,141,70,323),(4,140,71,322),(5,139,72,321),(6,138,73,320),(7,137,74,319),(8,136,75,318),(9,150,61,317),(10,149,62,316),(11,148,63,330),(12,147,64,329),(13,146,65,328),(14,145,66,327),(15,144,67,326),(16,266,203,168),(17,265,204,167),(18,264,205,166),(19,263,206,180),(20,262,207,179),(21,261,208,178),(22,260,209,177),(23,259,210,176),(24,258,196,175),(25,257,197,174),(26,256,198,173),(27,270,199,172),(28,269,200,171),(29,268,201,170),(30,267,202,169),(31,295,342,94),(32,294,343,93),(33,293,344,92),(34,292,345,91),(35,291,331,105),(36,290,332,104),(37,289,333,103),(38,288,334,102),(39,287,335,101),(40,286,336,100),(41,300,337,99),(42,299,338,98),(43,298,339,97),(44,297,340,96),(45,296,341,95),(46,184,349,308),(47,183,350,307),(48,182,351,306),(49,181,352,305),(50,195,353,304),(51,194,354,303),(52,193,355,302),(53,192,356,301),(54,191,357,315),(55,190,358,314),(56,189,359,313),(57,188,360,312),(58,187,346,311),(59,186,347,310),(60,185,348,309),(76,212,272,235),(77,211,273,234),(78,225,274,233),(79,224,275,232),(80,223,276,231),(81,222,277,230),(82,221,278,229),(83,220,279,228),(84,219,280,227),(85,218,281,226),(86,217,282,240),(87,216,283,239),(88,215,284,238),(89,214,285,237),(90,213,271,236),(106,160,129,250),(107,159,130,249),(108,158,131,248),(109,157,132,247),(110,156,133,246),(111,155,134,245),(112,154,135,244),(113,153,121,243),(114,152,122,242),(115,151,123,241),(116,165,124,255),(117,164,125,254),(118,163,126,253),(119,162,127,252),(120,161,128,251)]])`

93 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 4C 5A 5B 6A 6B 6C 6D 10A 10B 12A ··· 12H 15A ··· 15P 20A 20B 20C 20D 30A ··· 30P 60A ··· 60AF order 1 2 3 3 3 3 4 4 4 5 5 6 6 6 6 10 10 12 ··· 12 15 ··· 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 2 2 2 2 2 90 90 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

93 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + + + - + - + - image C1 C2 C2 S3 Q8 D5 D6 D10 Dic6 D15 Dic10 D30 Dic30 kernel C12.D15 C3⋊Dic15 C3×C60 C60 C3×C15 C3×C12 C30 C3×C6 C15 C12 C32 C6 C3 # reps 1 2 1 4 1 2 4 2 8 16 4 16 32

Matrix representation of C12.D15 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 20 2 0 0 32 49
,
 0 1 0 0 60 60 0 0 0 0 14 56 0 0 42 33
,
 17 25 0 0 8 44 0 0 0 0 25 9 0 0 5 36
`G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,20,32,0,0,2,49],[0,60,0,0,1,60,0,0,0,0,14,42,0,0,56,33],[17,8,0,0,25,44,0,0,0,0,25,5,0,0,9,36] >;`

C12.D15 in GAP, Magma, Sage, TeX

`C_{12}.D_{15}`
`% in TeX`

`G:=Group("C12.D15");`
`// GroupNames label`

`G:=SmallGroup(360,110);`
`// by ID`

`G=gap.SmallGroup(360,110);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,73,31,387,1444,10373]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^15=1,c^2=a^6,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;`
`// generators/relations`

׿
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