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G = C36.Q8order 288 = 25·32

1st non-split extension by C36 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C18.6D8, C36.1Q8, C18.3Q16, C12.1Dic6, C4.1Dic18, C9⋊C81C4, C4⋊C4.1D9, C91(C2.D8), C12.1(C4×S3), C36.1(C2×C4), C4.11(C4×D9), C18.2(C4⋊C4), C2.1(D4⋊D9), (C2×C18).29D4, (C2×C4).35D18, (C2×C12).37D6, C4⋊Dic9.7C2, C3.(C6.Q16), C6.13(D4⋊S3), C6.6(C3⋊Q16), C2.1(C9⋊Q16), C2.3(Dic9⋊C4), (C2×C36).15C22, C6.10(Dic3⋊C4), C22.12(C9⋊D4), (C2×C9⋊C8).1C2, (C9×C4⋊C4).1C2, (C3×C4⋊C4).1S3, (C2×C6).67(C3⋊D4), SmallGroup(288,14)

Series: Derived Chief Lower central Upper central

C1C36 — C36.Q8
C1C3C9C18C2×C18C2×C36C2×C9⋊C8 — C36.Q8
C9C18C36 — C36.Q8
C1C22C2×C4C4⋊C4

Generators and relations for C36.Q8
 G = < a,b,c | a36=b4=1, c2=a9b2, bab-1=a19, cac-1=a17, cbc-1=a9b-1 >

4C4
36C4
2C2×C4
9C8
9C8
18C2×C4
4C12
12Dic3
9C4⋊C4
9C2×C8
2C2×C12
3C3⋊C8
3C3⋊C8
6C2×Dic3
4C36
4Dic9
9C2.D8
3C4⋊Dic3
3C2×C3⋊C8
2C2×C36
2C2×Dic9
3C6.Q16

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

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

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

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

54 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F6A6B6C8A8B8C8D9A9B9C12A···12F18A···18I36A···36R
order12223444444666888899912···1218···1836···36
size1111222443636222181818182224···42···24···4

54 irreducible representations

dim11111222222222222224444
type+++++-+++-+-+-+-+-
imageC1C2C2C2C4S3Q8D4D6D8Q16D9Dic6C4×S3C3⋊D4D18Dic18C4×D9C9⋊D4D4⋊S3C3⋊Q16D4⋊D9C9⋊Q16
kernelC36.Q8C2×C9⋊C8C4⋊Dic9C9×C4⋊C4C9⋊C8C3×C4⋊C4C36C2×C18C2×C12C18C18C4⋊C4C12C12C2×C6C2×C4C4C4C22C6C6C2C2
# reps11114111122322236661133

Matrix representation of C36.Q8 in GL6(𝔽73)

010000
72720000
00287000
0033100
0000171
0000172
,
7140000
59660000
0072000
0007200
00003241
00001641
,
5520000
20180000
00704200
0045300
00003241
0000160

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,28,3,0,0,0,0,70,31,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[7,59,0,0,0,0,14,66,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,32,16,0,0,0,0,41,41],[55,20,0,0,0,0,2,18,0,0,0,0,0,0,70,45,0,0,0,0,42,3,0,0,0,0,0,0,32,16,0,0,0,0,41,0] >;

C36.Q8 in GAP, Magma, Sage, TeX

C_{36}.Q_8
% in TeX

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

G:=SmallGroup(288,14);
// by ID

G=gap.SmallGroup(288,14);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,346,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=b^4=1,c^2=a^9*b^2,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^9*b^-1>;
// generators/relations

Export

Subgroup lattice of C36.Q8 in TeX

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