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## G = C36.6Q8order 288 = 25·32

### 3rd non-split extension by C36 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C36.6Q8
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — Dic9⋊C4 — C36.6Q8
 Lower central C9 — C2×C18 — C36.6Q8
 Upper central C1 — C22 — C42

Generators and relations for C36.6Q8
G = < a,b,c | a36=b4=1, c2=a18b2, ab=ba, cac-1=a-1, cbc-1=a18b-1 >

Subgroups: 292 in 84 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×6], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], C9, Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C42, C4⋊C4 [×6], C18, C18 [×2], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C42.C2, Dic9 [×4], C36 [×2], C36 [×2], C2×C18, Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4×C12, C2×Dic9 [×4], C2×C36, C2×C36 [×2], C12.6Q8, Dic9⋊C4 [×4], C4⋊Dic9 [×2], C4×C36, C36.6Q8
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, C4○D12 [×2], Dic18 [×2], C22×D9, C12.6Q8, C2×Dic18, D365C2 [×2], C36.6Q8

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

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4F 4G 4H 4I 4J 6A 6B 6C 9A 9B 9C 12A ··· 12L 18A ··· 18I 36A ··· 36AJ order 1 2 2 2 3 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 36 36 36 36 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - + - image C1 C2 C2 C2 S3 Q8 D6 C4○D4 D9 Dic6 D18 C4○D12 Dic18 D36⋊5C2 kernel C36.6Q8 Dic9⋊C4 C4⋊Dic9 C4×C36 C4×C12 C36 C2×C12 C18 C42 C12 C2×C4 C6 C4 C2 # reps 1 4 2 1 1 2 3 4 3 4 9 8 12 24

Matrix representation of C36.6Q8 in GL4(𝔽37) generated by

 0 6 0 0 6 0 0 0 0 0 17 31 0 0 6 11
,
 6 0 0 0 0 6 0 0 0 0 5 27 0 0 10 32
,
 21 12 0 0 25 16 0 0 0 0 36 29 0 0 28 1
`G:=sub<GL(4,GF(37))| [0,6,0,0,6,0,0,0,0,0,17,6,0,0,31,11],[6,0,0,0,0,6,0,0,0,0,5,10,0,0,27,32],[21,25,0,0,12,16,0,0,0,0,36,28,0,0,29,1] >;`

C36.6Q8 in GAP, Magma, Sage, TeX

`C_{36}._6Q_8`
`% in TeX`

`G:=Group("C36.6Q8");`
`// GroupNames label`

`G:=SmallGroup(288,80);`
`// by ID`

`G=gap.SmallGroup(288,80);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,254,100,6725,292,9414]);`
`// Polycyclic`

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

׿
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