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## G = C36.3Q8order 288 = 25·32

### 3rd non-split extension by C36 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C36.3Q8
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C4×Dic9 — C36.3Q8
 Lower central C9 — C2×C18 — C36.3Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C36.3Q8
G = < a,b,c | a36=b4=1, c2=b2, bab-1=a19, cac-1=a17, cbc-1=a18b-1 >

Subgroups: 292 in 84 conjugacy classes, 44 normal (26 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×6], C22, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], C9, Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C42, C4⋊C4, C4⋊C4 [×5], C18 [×3], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C42.C2, Dic9 [×4], C36 [×2], C36 [×2], C2×C18, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3 [×3], C3×C4⋊C4, C2×Dic9 [×2], C2×Dic9 [×2], C2×C36, C2×C36 [×2], C4.Dic6, C4×Dic9, Dic9⋊C4 [×2], C4⋊Dic9, C4⋊Dic9 [×2], C9×C4⋊C4, C36.3Q8
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], C2×Q8, C4○D4 [×2], D9, Dic6 [×2], C22×S3, C42.C2, D18 [×3], C2×Dic6, D42S3, Q83S3, Dic18 [×2], C22×D9, C4.Dic6, C2×Dic18, D42D9, Q83D9, C36.3Q8

Smallest permutation representation of C36.3Q8
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)
(1 246 40 131)(2 229 41 114)(3 248 42 133)(4 231 43 116)(5 250 44 135)(6 233 45 118)(7 252 46 137)(8 235 47 120)(9 218 48 139)(10 237 49 122)(11 220 50 141)(12 239 51 124)(13 222 52 143)(14 241 53 126)(15 224 54 109)(16 243 55 128)(17 226 56 111)(18 245 57 130)(19 228 58 113)(20 247 59 132)(21 230 60 115)(22 249 61 134)(23 232 62 117)(24 251 63 136)(25 234 64 119)(26 217 65 138)(27 236 66 121)(28 219 67 140)(29 238 68 123)(30 221 69 142)(31 240 70 125)(32 223 71 144)(33 242 72 127)(34 225 37 110)(35 244 38 129)(36 227 39 112)(73 200 145 286)(74 183 146 269)(75 202 147 288)(76 185 148 271)(77 204 149 254)(78 187 150 273)(79 206 151 256)(80 189 152 275)(81 208 153 258)(82 191 154 277)(83 210 155 260)(84 193 156 279)(85 212 157 262)(86 195 158 281)(87 214 159 264)(88 197 160 283)(89 216 161 266)(90 199 162 285)(91 182 163 268)(92 201 164 287)(93 184 165 270)(94 203 166 253)(95 186 167 272)(96 205 168 255)(97 188 169 274)(98 207 170 257)(99 190 171 276)(100 209 172 259)(101 192 173 278)(102 211 174 261)(103 194 175 280)(104 213 176 263)(105 196 177 282)(106 215 178 265)(107 198 179 284)(108 181 180 267)
(1 162 40 90)(2 179 41 107)(3 160 42 88)(4 177 43 105)(5 158 44 86)(6 175 45 103)(7 156 46 84)(8 173 47 101)(9 154 48 82)(10 171 49 99)(11 152 50 80)(12 169 51 97)(13 150 52 78)(14 167 53 95)(15 148 54 76)(16 165 55 93)(17 146 56 74)(18 163 57 91)(19 180 58 108)(20 161 59 89)(21 178 60 106)(22 159 61 87)(23 176 62 104)(24 157 63 85)(25 174 64 102)(26 155 65 83)(27 172 66 100)(28 153 67 81)(29 170 68 98)(30 151 69 79)(31 168 70 96)(32 149 71 77)(33 166 72 94)(34 147 37 75)(35 164 38 92)(36 145 39 73)(109 253 224 203)(110 270 225 184)(111 287 226 201)(112 268 227 182)(113 285 228 199)(114 266 229 216)(115 283 230 197)(116 264 231 214)(117 281 232 195)(118 262 233 212)(119 279 234 193)(120 260 235 210)(121 277 236 191)(122 258 237 208)(123 275 238 189)(124 256 239 206)(125 273 240 187)(126 254 241 204)(127 271 242 185)(128 288 243 202)(129 269 244 183)(130 286 245 200)(131 267 246 181)(132 284 247 198)(133 265 248 215)(134 282 249 196)(135 263 250 213)(136 280 251 194)(137 261 252 211)(138 278 217 192)(139 259 218 209)(140 276 219 190)(141 257 220 207)(142 274 221 188)(143 255 222 205)(144 272 223 186)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 4 4 18 18 18 18 36 36 2 2 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + - + - - + - + image C1 C2 C2 C2 C2 S3 Q8 D6 C4○D4 D9 Dic6 D18 Dic18 D4⋊2S3 Q8⋊3S3 D4⋊2D9 Q8⋊3D9 kernel C36.3Q8 C4×Dic9 Dic9⋊C4 C4⋊Dic9 C9×C4⋊C4 C3×C4⋊C4 C36 C2×C12 C18 C4⋊C4 C12 C2×C4 C4 C6 C6 C2 C2 # reps 1 1 2 3 1 1 2 3 4 3 4 9 12 1 1 3 3

Matrix representation of C36.3Q8 in GL4(𝔽37) generated by

 30 0 0 0 0 21 0 0 0 0 11 27 0 0 27 26
,
 6 0 0 0 0 31 0 0 0 0 0 1 0 0 36 0
,
 0 1 0 0 36 0 0 0 0 0 6 0 0 0 0 6
`G:=sub<GL(4,GF(37))| [30,0,0,0,0,21,0,0,0,0,11,27,0,0,27,26],[6,0,0,0,0,31,0,0,0,0,0,36,0,0,1,0],[0,36,0,0,1,0,0,0,0,0,6,0,0,0,0,6] >;`

C36.3Q8 in GAP, Magma, Sage, TeX

`C_{36}._3Q_8`
`% in TeX`

`G:=Group("C36.3Q8");`
`// GroupNames label`

`G:=SmallGroup(288,100);`
`// by ID`

`G=gap.SmallGroup(288,100);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,120,590,219,58,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=b^4=1,c^2=b^2,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^18*b^-1>;`
`// generators/relations`

׿
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