metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊2Q8, C8⋊4Dic6, Dic3.2D8, Dic3.2Q16, C3⋊C8⋊6Q8, C4⋊C4.43D6, C3⋊2(C8⋊2Q8), C2.12(S3×D8), C6.27(C2×D8), C12⋊Q8.7C2, C4.26(S3×Q8), C2.D8.4S3, (C2×C8).227D6, C6.16(C4⋊Q8), C6.21(C2×Q16), C12.17(C2×Q8), C2.12(S3×Q16), C2.11(C12⋊Q8), (C8×Dic3).2C2, C24⋊1C4.14C2, C4.23(C2×Dic6), C6.Q16.7C2, (C2×C24).79C22, C22.223(S3×D4), (C2×C12).290C23, (C2×Dic3).100D4, C4⋊Dic3.116C22, (C4×Dic3).233C22, (C3×C2.D8).5C2, (C2×C6).295(C2×D4), (C3×C4⋊C4).83C22, (C2×C3⋊C8).231C22, (C2×C4).393(C22×S3), SmallGroup(192,433)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊2Q8
G = < a,b,c | a24=b4=1, c2=b2, bab-1=a7, cac-1=a17, cbc-1=b-1 >
Subgroups: 272 in 98 conjugacy classes, 47 normal (27 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4×C8, C2.D8, C2.D8, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C8⋊2Q8, C6.Q16, C8×Dic3, C24⋊1C4, C3×C2.D8, C12⋊Q8, C24⋊2Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, D8, Q16, C2×D4, C2×Q8, Dic6, C22×S3, C4⋊Q8, C2×D8, C2×Q16, C2×Dic6, S3×D4, S3×Q8, C8⋊2Q8, C12⋊Q8, S3×D8, S3×Q16, C24⋊2Q8
►Regular action on 192 points
Generators in S192
(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)
(1 25 134 68)(2 32 135 51)(3 39 136 58)(4 46 137 65)(5 29 138 72)(6 36 139 55)(7 43 140 62)(8 26 141 69)(9 33 142 52)(10 40 143 59)(11 47 144 66)(12 30 121 49)(13 37 122 56)(14 44 123 63)(15 27 124 70)(16 34 125 53)(17 41 126 60)(18 48 127 67)(19 31 128 50)(20 38 129 57)(21 45 130 64)(22 28 131 71)(23 35 132 54)(24 42 133 61)(73 170 150 99)(74 177 151 106)(75 184 152 113)(76 191 153 120)(77 174 154 103)(78 181 155 110)(79 188 156 117)(80 171 157 100)(81 178 158 107)(82 185 159 114)(83 192 160 97)(84 175 161 104)(85 182 162 111)(86 189 163 118)(87 172 164 101)(88 179 165 108)(89 186 166 115)(90 169 167 98)(91 176 168 105)(92 183 145 112)(93 190 146 119)(94 173 147 102)(95 180 148 109)(96 187 149 116)
(1 176 134 105)(2 169 135 98)(3 186 136 115)(4 179 137 108)(5 172 138 101)(6 189 139 118)(7 182 140 111)(8 175 141 104)(9 192 142 97)(10 185 143 114)(11 178 144 107)(12 171 121 100)(13 188 122 117)(14 181 123 110)(15 174 124 103)(16 191 125 120)(17 184 126 113)(18 177 127 106)(19 170 128 99)(20 187 129 116)(21 180 130 109)(22 173 131 102)(23 190 132 119)(24 183 133 112)(25 91 68 168)(26 84 69 161)(27 77 70 154)(28 94 71 147)(29 87 72 164)(30 80 49 157)(31 73 50 150)(32 90 51 167)(33 83 52 160)(34 76 53 153)(35 93 54 146)(36 86 55 163)(37 79 56 156)(38 96 57 149)(39 89 58 166)(40 82 59 159)(41 75 60 152)(42 92 61 145)(43 85 62 162)(44 78 63 155)(45 95 64 148)(46 88 65 165)(47 81 66 158)(48 74 67 151)
G:=sub<Sym(192)| (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), (1,25,134,68)(2,32,135,51)(3,39,136,58)(4,46,137,65)(5,29,138,72)(6,36,139,55)(7,43,140,62)(8,26,141,69)(9,33,142,52)(10,40,143,59)(11,47,144,66)(12,30,121,49)(13,37,122,56)(14,44,123,63)(15,27,124,70)(16,34,125,53)(17,41,126,60)(18,48,127,67)(19,31,128,50)(20,38,129,57)(21,45,130,64)(22,28,131,71)(23,35,132,54)(24,42,133,61)(73,170,150,99)(74,177,151,106)(75,184,152,113)(76,191,153,120)(77,174,154,103)(78,181,155,110)(79,188,156,117)(80,171,157,100)(81,178,158,107)(82,185,159,114)(83,192,160,97)(84,175,161,104)(85,182,162,111)(86,189,163,118)(87,172,164,101)(88,179,165,108)(89,186,166,115)(90,169,167,98)(91,176,168,105)(92,183,145,112)(93,190,146,119)(94,173,147,102)(95,180,148,109)(96,187,149,116), (1,176,134,105)(2,169,135,98)(3,186,136,115)(4,179,137,108)(5,172,138,101)(6,189,139,118)(7,182,140,111)(8,175,141,104)(9,192,142,97)(10,185,143,114)(11,178,144,107)(12,171,121,100)(13,188,122,117)(14,181,123,110)(15,174,124,103)(16,191,125,120)(17,184,126,113)(18,177,127,106)(19,170,128,99)(20,187,129,116)(21,180,130,109)(22,173,131,102)(23,190,132,119)(24,183,133,112)(25,91,68,168)(26,84,69,161)(27,77,70,154)(28,94,71,147)(29,87,72,164)(30,80,49,157)(31,73,50,150)(32,90,51,167)(33,83,52,160)(34,76,53,153)(35,93,54,146)(36,86,55,163)(37,79,56,156)(38,96,57,149)(39,89,58,166)(40,82,59,159)(41,75,60,152)(42,92,61,145)(43,85,62,162)(44,78,63,155)(45,95,64,148)(46,88,65,165)(47,81,66,158)(48,74,67,151)>;
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), (1,25,134,68)(2,32,135,51)(3,39,136,58)(4,46,137,65)(5,29,138,72)(6,36,139,55)(7,43,140,62)(8,26,141,69)(9,33,142,52)(10,40,143,59)(11,47,144,66)(12,30,121,49)(13,37,122,56)(14,44,123,63)(15,27,124,70)(16,34,125,53)(17,41,126,60)(18,48,127,67)(19,31,128,50)(20,38,129,57)(21,45,130,64)(22,28,131,71)(23,35,132,54)(24,42,133,61)(73,170,150,99)(74,177,151,106)(75,184,152,113)(76,191,153,120)(77,174,154,103)(78,181,155,110)(79,188,156,117)(80,171,157,100)(81,178,158,107)(82,185,159,114)(83,192,160,97)(84,175,161,104)(85,182,162,111)(86,189,163,118)(87,172,164,101)(88,179,165,108)(89,186,166,115)(90,169,167,98)(91,176,168,105)(92,183,145,112)(93,190,146,119)(94,173,147,102)(95,180,148,109)(96,187,149,116), (1,176,134,105)(2,169,135,98)(3,186,136,115)(4,179,137,108)(5,172,138,101)(6,189,139,118)(7,182,140,111)(8,175,141,104)(9,192,142,97)(10,185,143,114)(11,178,144,107)(12,171,121,100)(13,188,122,117)(14,181,123,110)(15,174,124,103)(16,191,125,120)(17,184,126,113)(18,177,127,106)(19,170,128,99)(20,187,129,116)(21,180,130,109)(22,173,131,102)(23,190,132,119)(24,183,133,112)(25,91,68,168)(26,84,69,161)(27,77,70,154)(28,94,71,147)(29,87,72,164)(30,80,49,157)(31,73,50,150)(32,90,51,167)(33,83,52,160)(34,76,53,153)(35,93,54,146)(36,86,55,163)(37,79,56,156)(38,96,57,149)(39,89,58,166)(40,82,59,159)(41,75,60,152)(42,92,61,145)(43,85,62,162)(44,78,63,155)(45,95,64,148)(46,88,65,165)(47,81,66,158)(48,74,67,151) );
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)], [(1,25,134,68),(2,32,135,51),(3,39,136,58),(4,46,137,65),(5,29,138,72),(6,36,139,55),(7,43,140,62),(8,26,141,69),(9,33,142,52),(10,40,143,59),(11,47,144,66),(12,30,121,49),(13,37,122,56),(14,44,123,63),(15,27,124,70),(16,34,125,53),(17,41,126,60),(18,48,127,67),(19,31,128,50),(20,38,129,57),(21,45,130,64),(22,28,131,71),(23,35,132,54),(24,42,133,61),(73,170,150,99),(74,177,151,106),(75,184,152,113),(76,191,153,120),(77,174,154,103),(78,181,155,110),(79,188,156,117),(80,171,157,100),(81,178,158,107),(82,185,159,114),(83,192,160,97),(84,175,161,104),(85,182,162,111),(86,189,163,118),(87,172,164,101),(88,179,165,108),(89,186,166,115),(90,169,167,98),(91,176,168,105),(92,183,145,112),(93,190,146,119),(94,173,147,102),(95,180,148,109),(96,187,149,116)], [(1,176,134,105),(2,169,135,98),(3,186,136,115),(4,179,137,108),(5,172,138,101),(6,189,139,118),(7,182,140,111),(8,175,141,104),(9,192,142,97),(10,185,143,114),(11,178,144,107),(12,171,121,100),(13,188,122,117),(14,181,123,110),(15,174,124,103),(16,191,125,120),(17,184,126,113),(18,177,127,106),(19,170,128,99),(20,187,129,116),(21,180,130,109),(22,173,131,102),(23,190,132,119),(24,183,133,112),(25,91,68,168),(26,84,69,161),(27,77,70,154),(28,94,71,147),(29,87,72,164),(30,80,49,157),(31,73,50,150),(32,90,51,167),(33,83,52,160),(34,76,53,153),(35,93,54,146),(36,86,55,163),(37,79,56,156),(38,96,57,149),(39,89,58,166),(40,82,59,159),(41,75,60,152),(42,92,61,145),(43,85,62,162),(44,78,63,155),(45,95,64,148),(46,88,65,165),(47,81,66,158),(48,74,67,151)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | + | + | - | - | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | Q8 | D4 | D6 | D6 | D8 | Q16 | Dic6 | S3×Q8 | S3×D4 | S3×D8 | S3×Q16 |
| kernel | C24⋊2Q8 | C6.Q16 | C8×Dic3 | C24⋊1C4 | C3×C2.D8 | C12⋊Q8 | C2.D8 | C3⋊C8 | C24 | C2×Dic3 | C4⋊C4 | C2×C8 | Dic3 | Dic3 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C24⋊2Q8 ►in GL6(𝔽73)
| 41 | 32 | 0 | 0 | 0 | 0 |
| 57 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 42 | 0 | 0 | 0 | 0 |
| 61 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 30 |
| 0 | 0 | 0 | 0 | 30 | 11 |
G:=sub<GL(6,GF(73))| [41,57,0,0,0,0,32,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,61,0,0,0,0,42,33,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,62,30,0,0,0,0,30,11] >;
C24⋊2Q8 in GAP, Magma, Sage, TeX
C_{24}\rtimes_2Q_8 % in TeX
G:=Group("C24:2Q8"); // GroupNames label
G:=SmallGroup(192,433);
// by ID
G=gap.SmallGroup(192,433);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,254,219,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=b^2,b*a*b^-1=a^7,c*a*c^-1=a^17,c*b*c^-1=b^-1>;
// generators/relations