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## G = C36.45D4order 288 = 25·32

### 1st non-split extension by C36 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C36.45D4
 Chief series C1 — C3 — C9 — C18 — C36 — C2×C36 — C4⋊Dic9 — C36.45D4
 Lower central C9 — C18 — C36 — C36.45D4
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C36.45D4
G = < a,b,c | a36=b4=1, c2=a18, bab-1=cac-1=a-1, cbc-1=a9b-1 >

Subgroups: 280 in 63 conjugacy classes, 32 normal (30 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C9, Dic3 [×3], C12 [×2], C2×C6, C4⋊C4, C2×C8, C2×Q8, C18 [×3], C24, Dic6 [×3], C2×Dic3 [×2], C2×C12, Q8⋊C4, Dic9 [×3], C36 [×2], C2×C18, C4⋊Dic3, C2×C24, C2×Dic6, C72, Dic18 [×2], Dic18, C2×Dic9 [×2], C2×C36, C2.Dic12, C4⋊Dic9, C2×C72, C2×Dic18, C36.45D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, SD16, Q16, D9, C4×S3, D12, C3⋊D4, Q8⋊C4, D18, C24⋊C2, Dic12, D6⋊C4, C4×D9, D36, C9⋊D4, C2.Dic12, Dic36, C72⋊C2, D18⋊C4, C36.45D4

Smallest permutation representation of C36.45D4
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 67 158 237)(2 66 159 236)(3 65 160 235)(4 64 161 234)(5 63 162 233)(6 62 163 232)(7 61 164 231)(8 60 165 230)(9 59 166 229)(10 58 167 228)(11 57 168 227)(12 56 169 226)(13 55 170 225)(14 54 171 224)(15 53 172 223)(16 52 173 222)(17 51 174 221)(18 50 175 220)(19 49 176 219)(20 48 177 218)(21 47 178 217)(22 46 179 252)(23 45 180 251)(24 44 145 250)(25 43 146 249)(26 42 147 248)(27 41 148 247)(28 40 149 246)(29 39 150 245)(30 38 151 244)(31 37 152 243)(32 72 153 242)(33 71 154 241)(34 70 155 240)(35 69 156 239)(36 68 157 238)(73 255 136 210)(74 254 137 209)(75 253 138 208)(76 288 139 207)(77 287 140 206)(78 286 141 205)(79 285 142 204)(80 284 143 203)(81 283 144 202)(82 282 109 201)(83 281 110 200)(84 280 111 199)(85 279 112 198)(86 278 113 197)(87 277 114 196)(88 276 115 195)(89 275 116 194)(90 274 117 193)(91 273 118 192)(92 272 119 191)(93 271 120 190)(94 270 121 189)(95 269 122 188)(96 268 123 187)(97 267 124 186)(98 266 125 185)(99 265 126 184)(100 264 127 183)(101 263 128 182)(102 262 129 181)(103 261 130 216)(104 260 131 215)(105 259 132 214)(106 258 133 213)(107 257 134 212)(108 256 135 211)
(1 140 19 122)(2 139 20 121)(3 138 21 120)(4 137 22 119)(5 136 23 118)(6 135 24 117)(7 134 25 116)(8 133 26 115)(9 132 27 114)(10 131 28 113)(11 130 29 112)(12 129 30 111)(13 128 31 110)(14 127 32 109)(15 126 33 144)(16 125 34 143)(17 124 35 142)(18 123 36 141)(37 272 55 254)(38 271 56 253)(39 270 57 288)(40 269 58 287)(41 268 59 286)(42 267 60 285)(43 266 61 284)(44 265 62 283)(45 264 63 282)(46 263 64 281)(47 262 65 280)(48 261 66 279)(49 260 67 278)(50 259 68 277)(51 258 69 276)(52 257 70 275)(53 256 71 274)(54 255 72 273)(73 180 91 162)(74 179 92 161)(75 178 93 160)(76 177 94 159)(77 176 95 158)(78 175 96 157)(79 174 97 156)(80 173 98 155)(81 172 99 154)(82 171 100 153)(83 170 101 152)(84 169 102 151)(85 168 103 150)(86 167 104 149)(87 166 105 148)(88 165 106 147)(89 164 107 146)(90 163 108 145)(181 235 199 217)(182 234 200 252)(183 233 201 251)(184 232 202 250)(185 231 203 249)(186 230 204 248)(187 229 205 247)(188 228 206 246)(189 227 207 245)(190 226 208 244)(191 225 209 243)(192 224 210 242)(193 223 211 241)(194 222 212 240)(195 221 213 239)(196 220 214 238)(197 219 215 237)(198 218 216 236)```

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 24A ··· 24H 36A ··· 36L 72A ··· 72X order 1 2 2 2 3 4 4 4 4 4 4 6 6 6 8 8 8 8 9 9 9 12 12 12 12 18 ··· 18 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 1 1 2 2 2 36 36 36 36 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + + + - + - image C1 C2 C2 C2 C4 S3 D4 D4 D6 SD16 Q16 D9 C4×S3 C3⋊D4 D12 D18 C24⋊C2 Dic12 C4×D9 C9⋊D4 D36 Dic36 C72⋊C2 kernel C36.45D4 C4⋊Dic9 C2×C72 C2×Dic18 Dic18 C2×C24 C36 C2×C18 C2×C12 C18 C18 C2×C8 C12 C12 C2×C6 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 3 2 2 2 3 4 4 6 6 6 12 12

Matrix representation of C36.45D4 in GL4(𝔽73) generated by

 45 3 0 0 70 42 0 0 0 0 29 54 0 0 19 48
,
 0 27 0 0 27 0 0 0 0 0 20 2 0 0 55 53
,
 0 1 0 0 1 0 0 0 0 0 61 10 0 0 22 12
`G:=sub<GL(4,GF(73))| [45,70,0,0,3,42,0,0,0,0,29,19,0,0,54,48],[0,27,0,0,27,0,0,0,0,0,20,55,0,0,2,53],[0,1,0,0,1,0,0,0,0,0,61,22,0,0,10,12] >;`

C36.45D4 in GAP, Magma, Sage, TeX

`C_{36}._{45}D_4`
`% in TeX`

`G:=Group("C36.45D4");`
`// GroupNames label`

`G:=SmallGroup(288,24);`
`// by ID`

`G=gap.SmallGroup(288,24);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,85,92,422,100,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=b^4=1,c^2=a^18,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;`
`// generators/relations`

׿
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