metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊7Q8, C8⋊6Dic6, C42.12D6, Dic3.1M4(2), C6.7(C4×Q8), C3⋊2(C8⋊4Q8), C8⋊C4.5S3, (C2×C8).154D6, C4⋊Dic3.8C4, C12.81(C2×Q8), C12⋊C8.5C2, C2.9(C4×Dic6), C6.20(C8○D4), Dic3⋊C4.4C4, (C4×Dic6).4C2, (C2×Dic6).7C4, C24⋊C4.11C2, C4.46(C2×Dic6), C2.8(S3×M4(2)), Dic3⋊C8.15C2, C2.6(D12.C4), (C4×C12).12C22, (C8×Dic3).16C2, C6.14(C2×M4(2)), C4.128(C4○D12), C12.244(C4○D4), (C2×C24).267C22, (C2×C12).809C23, (C4×Dic3).180C22, (C2×C4).28(C4×S3), C22.97(S3×C2×C4), (C3×C8⋊C4).6C2, (C2×C12).36(C2×C4), (C2×C3⋊C8).293C22, (C2×C6).64(C22×C4), (C2×C4).751(C22×S3), (C2×Dic3).13(C2×C4), SmallGroup(192,260)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊Q8
G = < a,b,c | a24=b4=1, c2=b2, bab-1=a13, cac-1=a17, cbc-1=b-1 >
Subgroups: 184 in 94 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C24, C24, Dic6, C2×Dic3, C2×C12, C4×C8, C8⋊C4, C8⋊C4, C4⋊C8, C4×Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C8⋊4Q8, C12⋊C8, C8×Dic3, Dic3⋊C8, C24⋊C4, C3×C8⋊C4, C4×Dic6, C24⋊Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, M4(2), C22×C4, C2×Q8, C4○D4, Dic6, C4×S3, C22×S3, C4×Q8, C2×M4(2), C8○D4, C2×Dic6, S3×C2×C4, C4○D12, C8⋊4Q8, C4×Dic6, S3×M4(2), D12.C4, C24⋊Q8
►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)
(1 168 95 109)(2 157 96 98)(3 146 73 111)(4 159 74 100)(5 148 75 113)(6 161 76 102)(7 150 77 115)(8 163 78 104)(9 152 79 117)(10 165 80 106)(11 154 81 119)(12 167 82 108)(13 156 83 97)(14 145 84 110)(15 158 85 99)(16 147 86 112)(17 160 87 101)(18 149 88 114)(19 162 89 103)(20 151 90 116)(21 164 91 105)(22 153 92 118)(23 166 93 107)(24 155 94 120)(25 125 70 175)(26 138 71 188)(27 127 72 177)(28 140 49 190)(29 129 50 179)(30 142 51 192)(31 131 52 181)(32 144 53 170)(33 133 54 183)(34 122 55 172)(35 135 56 185)(36 124 57 174)(37 137 58 187)(38 126 59 176)(39 139 60 189)(40 128 61 178)(41 141 62 191)(42 130 63 180)(43 143 64 169)(44 132 65 182)(45 121 66 171)(46 134 67 184)(47 123 68 173)(48 136 69 186)
(1 133 95 183)(2 126 96 176)(3 143 73 169)(4 136 74 186)(5 129 75 179)(6 122 76 172)(7 139 77 189)(8 132 78 182)(9 125 79 175)(10 142 80 192)(11 135 81 185)(12 128 82 178)(13 121 83 171)(14 138 84 188)(15 131 85 181)(16 124 86 174)(17 141 87 191)(18 134 88 184)(19 127 89 177)(20 144 90 170)(21 137 91 187)(22 130 92 180)(23 123 93 173)(24 140 94 190)(25 117 70 152)(26 110 71 145)(27 103 72 162)(28 120 49 155)(29 113 50 148)(30 106 51 165)(31 99 52 158)(32 116 53 151)(33 109 54 168)(34 102 55 161)(35 119 56 154)(36 112 57 147)(37 105 58 164)(38 98 59 157)(39 115 60 150)(40 108 61 167)(41 101 62 160)(42 118 63 153)(43 111 64 146)(44 104 65 163)(45 97 66 156)(46 114 67 149)(47 107 68 166)(48 100 69 159)
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), (1,168,95,109)(2,157,96,98)(3,146,73,111)(4,159,74,100)(5,148,75,113)(6,161,76,102)(7,150,77,115)(8,163,78,104)(9,152,79,117)(10,165,80,106)(11,154,81,119)(12,167,82,108)(13,156,83,97)(14,145,84,110)(15,158,85,99)(16,147,86,112)(17,160,87,101)(18,149,88,114)(19,162,89,103)(20,151,90,116)(21,164,91,105)(22,153,92,118)(23,166,93,107)(24,155,94,120)(25,125,70,175)(26,138,71,188)(27,127,72,177)(28,140,49,190)(29,129,50,179)(30,142,51,192)(31,131,52,181)(32,144,53,170)(33,133,54,183)(34,122,55,172)(35,135,56,185)(36,124,57,174)(37,137,58,187)(38,126,59,176)(39,139,60,189)(40,128,61,178)(41,141,62,191)(42,130,63,180)(43,143,64,169)(44,132,65,182)(45,121,66,171)(46,134,67,184)(47,123,68,173)(48,136,69,186), (1,133,95,183)(2,126,96,176)(3,143,73,169)(4,136,74,186)(5,129,75,179)(6,122,76,172)(7,139,77,189)(8,132,78,182)(9,125,79,175)(10,142,80,192)(11,135,81,185)(12,128,82,178)(13,121,83,171)(14,138,84,188)(15,131,85,181)(16,124,86,174)(17,141,87,191)(18,134,88,184)(19,127,89,177)(20,144,90,170)(21,137,91,187)(22,130,92,180)(23,123,93,173)(24,140,94,190)(25,117,70,152)(26,110,71,145)(27,103,72,162)(28,120,49,155)(29,113,50,148)(30,106,51,165)(31,99,52,158)(32,116,53,151)(33,109,54,168)(34,102,55,161)(35,119,56,154)(36,112,57,147)(37,105,58,164)(38,98,59,157)(39,115,60,150)(40,108,61,167)(41,101,62,160)(42,118,63,153)(43,111,64,146)(44,104,65,163)(45,97,66,156)(46,114,67,149)(47,107,68,166)(48,100,69,159)>;
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), (1,168,95,109)(2,157,96,98)(3,146,73,111)(4,159,74,100)(5,148,75,113)(6,161,76,102)(7,150,77,115)(8,163,78,104)(9,152,79,117)(10,165,80,106)(11,154,81,119)(12,167,82,108)(13,156,83,97)(14,145,84,110)(15,158,85,99)(16,147,86,112)(17,160,87,101)(18,149,88,114)(19,162,89,103)(20,151,90,116)(21,164,91,105)(22,153,92,118)(23,166,93,107)(24,155,94,120)(25,125,70,175)(26,138,71,188)(27,127,72,177)(28,140,49,190)(29,129,50,179)(30,142,51,192)(31,131,52,181)(32,144,53,170)(33,133,54,183)(34,122,55,172)(35,135,56,185)(36,124,57,174)(37,137,58,187)(38,126,59,176)(39,139,60,189)(40,128,61,178)(41,141,62,191)(42,130,63,180)(43,143,64,169)(44,132,65,182)(45,121,66,171)(46,134,67,184)(47,123,68,173)(48,136,69,186), (1,133,95,183)(2,126,96,176)(3,143,73,169)(4,136,74,186)(5,129,75,179)(6,122,76,172)(7,139,77,189)(8,132,78,182)(9,125,79,175)(10,142,80,192)(11,135,81,185)(12,128,82,178)(13,121,83,171)(14,138,84,188)(15,131,85,181)(16,124,86,174)(17,141,87,191)(18,134,88,184)(19,127,89,177)(20,144,90,170)(21,137,91,187)(22,130,92,180)(23,123,93,173)(24,140,94,190)(25,117,70,152)(26,110,71,145)(27,103,72,162)(28,120,49,155)(29,113,50,148)(30,106,51,165)(31,99,52,158)(32,116,53,151)(33,109,54,168)(34,102,55,161)(35,119,56,154)(36,112,57,147)(37,105,58,164)(38,98,59,157)(39,115,60,150)(40,108,61,167)(41,101,62,160)(42,118,63,153)(43,111,64,146)(44,104,65,163)(45,97,66,156)(46,114,67,149)(47,107,68,166)(48,100,69,159) );
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)], [(1,168,95,109),(2,157,96,98),(3,146,73,111),(4,159,74,100),(5,148,75,113),(6,161,76,102),(7,150,77,115),(8,163,78,104),(9,152,79,117),(10,165,80,106),(11,154,81,119),(12,167,82,108),(13,156,83,97),(14,145,84,110),(15,158,85,99),(16,147,86,112),(17,160,87,101),(18,149,88,114),(19,162,89,103),(20,151,90,116),(21,164,91,105),(22,153,92,118),(23,166,93,107),(24,155,94,120),(25,125,70,175),(26,138,71,188),(27,127,72,177),(28,140,49,190),(29,129,50,179),(30,142,51,192),(31,131,52,181),(32,144,53,170),(33,133,54,183),(34,122,55,172),(35,135,56,185),(36,124,57,174),(37,137,58,187),(38,126,59,176),(39,139,60,189),(40,128,61,178),(41,141,62,191),(42,130,63,180),(43,143,64,169),(44,132,65,182),(45,121,66,171),(46,134,67,184),(47,123,68,173),(48,136,69,186)], [(1,133,95,183),(2,126,96,176),(3,143,73,169),(4,136,74,186),(5,129,75,179),(6,122,76,172),(7,139,77,189),(8,132,78,182),(9,125,79,175),(10,142,80,192),(11,135,81,185),(12,128,82,178),(13,121,83,171),(14,138,84,188),(15,131,85,181),(16,124,86,174),(17,141,87,191),(18,134,88,184),(19,127,89,177),(20,144,90,170),(21,137,91,187),(22,130,92,180),(23,123,93,173),(24,140,94,190),(25,117,70,152),(26,110,71,145),(27,103,72,162),(28,120,49,155),(29,113,50,148),(30,106,51,165),(31,99,52,158),(32,116,53,151),(33,109,54,168),(34,102,55,161),(35,119,56,154),(36,112,57,147),(37,105,58,164),(38,98,59,157),(39,115,60,150),(40,108,61,167),(41,101,62,160),(42,118,63,153),(43,111,64,146),(44,104,65,163),(45,97,66,156),(46,114,67,149),(47,107,68,166),(48,100,69,159)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Q8 | D6 | D6 | M4(2) | C4○D4 | Dic6 | C4×S3 | C8○D4 | C4○D12 | S3×M4(2) | D12.C4 |
| kernel | C24⋊Q8 | C12⋊C8 | C8×Dic3 | Dic3⋊C8 | C24⋊C4 | C3×C8⋊C4 | C4×Dic6 | Dic3⋊C4 | C4⋊Dic3 | C2×Dic6 | C8⋊C4 | C24 | C42 | C2×C8 | Dic3 | C12 | C8 | C2×C4 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C24⋊Q8 ►in GL4(𝔽73) generated by
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 15 | 72 |
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 56 | 64 |
| 0 | 0 | 32 | 17 |
| 11 | 48 | 0 | 0 |
| 37 | 62 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [72,1,0,0,72,0,0,0,0,0,1,15,0,0,3,72],[7,59,0,0,14,66,0,0,0,0,56,32,0,0,64,17],[11,37,0,0,48,62,0,0,0,0,72,0,0,0,0,72] >;
C24⋊Q8 in GAP, Magma, Sage, TeX
C_{24}\rtimes Q_8 % in TeX
G:=Group("C24:Q8"); // GroupNames label
G:=SmallGroup(192,260);
// by ID
G=gap.SmallGroup(192,260);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,387,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=b^2,b*a*b^-1=a^13,c*a*c^-1=a^17,c*b*c^-1=b^-1>;
// generators/relations