metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.13Q8, C8.12Dic6, C42.255D6, (C4×C8).10S3, C6.4(C4⋊Q8), (C4×C24).12C2, C6.1(C4○D8), (C2×C8).316D6, (C2×C4).59D12, C12.71(C2×Q8), C24⋊1C4.5C2, C8⋊Dic3.5C2, C3⋊1(C8.5Q8), C2.6(C4○D24), (C2×C12).349D4, C4.37(C2×Dic6), C2.8(C12⋊2Q8), C22.88(C2×D12), C4⋊Dic3.4C22, C12.6Q8.1C2, (C2×C12).719C23, (C2×C24).388C22, (C4×C12).305C22, (C2×C6).102(C2×D4), (C2×C4).662(C22×S3), SmallGroup(192,242)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.13Q8
G = < a,b,c | a24=b4=1, c2=a12b2, ab=ba, cac-1=a-1, cbc-1=a12b-1 >
Subgroups: 216 in 86 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C24, C2×Dic3, C2×C12, C2×C12, C4×C8, C4.Q8, C2.D8, C42.C2, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C8.5Q8, C8⋊Dic3, C24⋊1C4, C4×C24, C12.6Q8, C24.13Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, Dic6, D12, C22×S3, C4⋊Q8, C4○D8, C2×Dic6, C2×D12, C8.5Q8, C12⋊2Q8, C4○D24, C24.13Q8
►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 37 187 156)(2 38 188 157)(3 39 189 158)(4 40 190 159)(5 41 191 160)(6 42 192 161)(7 43 169 162)(8 44 170 163)(9 45 171 164)(10 46 172 165)(11 47 173 166)(12 48 174 167)(13 25 175 168)(14 26 176 145)(15 27 177 146)(16 28 178 147)(17 29 179 148)(18 30 180 149)(19 31 181 150)(20 32 182 151)(21 33 183 152)(22 34 184 153)(23 35 185 154)(24 36 186 155)(49 119 76 140)(50 120 77 141)(51 97 78 142)(52 98 79 143)(53 99 80 144)(54 100 81 121)(55 101 82 122)(56 102 83 123)(57 103 84 124)(58 104 85 125)(59 105 86 126)(60 106 87 127)(61 107 88 128)(62 108 89 129)(63 109 90 130)(64 110 91 131)(65 111 92 132)(66 112 93 133)(67 113 94 134)(68 114 95 135)(69 115 96 136)(70 116 73 137)(71 117 74 138)(72 118 75 139)
(1 80 175 65)(2 79 176 64)(3 78 177 63)(4 77 178 62)(5 76 179 61)(6 75 180 60)(7 74 181 59)(8 73 182 58)(9 96 183 57)(10 95 184 56)(11 94 185 55)(12 93 186 54)(13 92 187 53)(14 91 188 52)(15 90 189 51)(16 89 190 50)(17 88 191 49)(18 87 192 72)(19 86 169 71)(20 85 170 70)(21 84 171 69)(22 83 172 68)(23 82 173 67)(24 81 174 66)(25 99 156 132)(26 98 157 131)(27 97 158 130)(28 120 159 129)(29 119 160 128)(30 118 161 127)(31 117 162 126)(32 116 163 125)(33 115 164 124)(34 114 165 123)(35 113 166 122)(36 112 167 121)(37 111 168 144)(38 110 145 143)(39 109 146 142)(40 108 147 141)(41 107 148 140)(42 106 149 139)(43 105 150 138)(44 104 151 137)(45 103 152 136)(46 102 153 135)(47 101 154 134)(48 100 155 133)
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), (1,37,187,156)(2,38,188,157)(3,39,189,158)(4,40,190,159)(5,41,191,160)(6,42,192,161)(7,43,169,162)(8,44,170,163)(9,45,171,164)(10,46,172,165)(11,47,173,166)(12,48,174,167)(13,25,175,168)(14,26,176,145)(15,27,177,146)(16,28,178,147)(17,29,179,148)(18,30,180,149)(19,31,181,150)(20,32,182,151)(21,33,183,152)(22,34,184,153)(23,35,185,154)(24,36,186,155)(49,119,76,140)(50,120,77,141)(51,97,78,142)(52,98,79,143)(53,99,80,144)(54,100,81,121)(55,101,82,122)(56,102,83,123)(57,103,84,124)(58,104,85,125)(59,105,86,126)(60,106,87,127)(61,107,88,128)(62,108,89,129)(63,109,90,130)(64,110,91,131)(65,111,92,132)(66,112,93,133)(67,113,94,134)(68,114,95,135)(69,115,96,136)(70,116,73,137)(71,117,74,138)(72,118,75,139), (1,80,175,65)(2,79,176,64)(3,78,177,63)(4,77,178,62)(5,76,179,61)(6,75,180,60)(7,74,181,59)(8,73,182,58)(9,96,183,57)(10,95,184,56)(11,94,185,55)(12,93,186,54)(13,92,187,53)(14,91,188,52)(15,90,189,51)(16,89,190,50)(17,88,191,49)(18,87,192,72)(19,86,169,71)(20,85,170,70)(21,84,171,69)(22,83,172,68)(23,82,173,67)(24,81,174,66)(25,99,156,132)(26,98,157,131)(27,97,158,130)(28,120,159,129)(29,119,160,128)(30,118,161,127)(31,117,162,126)(32,116,163,125)(33,115,164,124)(34,114,165,123)(35,113,166,122)(36,112,167,121)(37,111,168,144)(38,110,145,143)(39,109,146,142)(40,108,147,141)(41,107,148,140)(42,106,149,139)(43,105,150,138)(44,104,151,137)(45,103,152,136)(46,102,153,135)(47,101,154,134)(48,100,155,133)>;
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), (1,37,187,156)(2,38,188,157)(3,39,189,158)(4,40,190,159)(5,41,191,160)(6,42,192,161)(7,43,169,162)(8,44,170,163)(9,45,171,164)(10,46,172,165)(11,47,173,166)(12,48,174,167)(13,25,175,168)(14,26,176,145)(15,27,177,146)(16,28,178,147)(17,29,179,148)(18,30,180,149)(19,31,181,150)(20,32,182,151)(21,33,183,152)(22,34,184,153)(23,35,185,154)(24,36,186,155)(49,119,76,140)(50,120,77,141)(51,97,78,142)(52,98,79,143)(53,99,80,144)(54,100,81,121)(55,101,82,122)(56,102,83,123)(57,103,84,124)(58,104,85,125)(59,105,86,126)(60,106,87,127)(61,107,88,128)(62,108,89,129)(63,109,90,130)(64,110,91,131)(65,111,92,132)(66,112,93,133)(67,113,94,134)(68,114,95,135)(69,115,96,136)(70,116,73,137)(71,117,74,138)(72,118,75,139), (1,80,175,65)(2,79,176,64)(3,78,177,63)(4,77,178,62)(5,76,179,61)(6,75,180,60)(7,74,181,59)(8,73,182,58)(9,96,183,57)(10,95,184,56)(11,94,185,55)(12,93,186,54)(13,92,187,53)(14,91,188,52)(15,90,189,51)(16,89,190,50)(17,88,191,49)(18,87,192,72)(19,86,169,71)(20,85,170,70)(21,84,171,69)(22,83,172,68)(23,82,173,67)(24,81,174,66)(25,99,156,132)(26,98,157,131)(27,97,158,130)(28,120,159,129)(29,119,160,128)(30,118,161,127)(31,117,162,126)(32,116,163,125)(33,115,164,124)(34,114,165,123)(35,113,166,122)(36,112,167,121)(37,111,168,144)(38,110,145,143)(39,109,146,142)(40,108,147,141)(41,107,148,140)(42,106,149,139)(43,105,150,138)(44,104,151,137)(45,103,152,136)(46,102,153,135)(47,101,154,134)(48,100,155,133) );
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)], [(1,37,187,156),(2,38,188,157),(3,39,189,158),(4,40,190,159),(5,41,191,160),(6,42,192,161),(7,43,169,162),(8,44,170,163),(9,45,171,164),(10,46,172,165),(11,47,173,166),(12,48,174,167),(13,25,175,168),(14,26,176,145),(15,27,177,146),(16,28,178,147),(17,29,179,148),(18,30,180,149),(19,31,181,150),(20,32,182,151),(21,33,183,152),(22,34,184,153),(23,35,185,154),(24,36,186,155),(49,119,76,140),(50,120,77,141),(51,97,78,142),(52,98,79,143),(53,99,80,144),(54,100,81,121),(55,101,82,122),(56,102,83,123),(57,103,84,124),(58,104,85,125),(59,105,86,126),(60,106,87,127),(61,107,88,128),(62,108,89,129),(63,109,90,130),(64,110,91,131),(65,111,92,132),(66,112,93,133),(67,113,94,134),(68,114,95,135),(69,115,96,136),(70,116,73,137),(71,117,74,138),(72,118,75,139)], [(1,80,175,65),(2,79,176,64),(3,78,177,63),(4,77,178,62),(5,76,179,61),(6,75,180,60),(7,74,181,59),(8,73,182,58),(9,96,183,57),(10,95,184,56),(11,94,185,55),(12,93,186,54),(13,92,187,53),(14,91,188,52),(15,90,189,51),(16,89,190,50),(17,88,191,49),(18,87,192,72),(19,86,169,71),(20,85,170,70),(21,84,171,69),(22,83,172,68),(23,82,173,67),(24,81,174,66),(25,99,156,132),(26,98,157,131),(27,97,158,130),(28,120,159,129),(29,119,160,128),(30,118,161,127),(31,117,162,126),(32,116,163,125),(33,115,164,124),(34,114,165,123),(35,113,166,122),(36,112,167,121),(37,111,168,144),(38,110,145,143),(39,109,146,142),(40,108,147,141),(41,107,148,140),(42,106,149,139),(43,105,150,138),(44,104,151,137),(45,103,152,136),(46,102,153,135),(47,101,154,134),(48,100,155,133)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | Dic6 | D12 | C4○D8 | C4○D24 |
| kernel | C24.13Q8 | C8⋊Dic3 | C24⋊1C4 | C4×C24 | C12.6Q8 | C4×C8 | C24 | C2×C12 | C42 | C2×C8 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 8 | 16 |
Matrix representation of C24.13Q8 ►in GL4(𝔽73) generated by
| 18 | 23 | 0 | 0 |
| 50 | 68 | 0 | 0 |
| 0 | 0 | 7 | 66 |
| 0 | 0 | 7 | 14 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 7 | 14 |
| 0 | 0 | 59 | 66 |
| 3 | 31 | 0 | 0 |
| 28 | 70 | 0 | 0 |
| 0 | 0 | 12 | 63 |
| 0 | 0 | 51 | 61 |
G:=sub<GL(4,GF(73))| [18,50,0,0,23,68,0,0,0,0,7,7,0,0,66,14],[27,0,0,0,0,27,0,0,0,0,7,59,0,0,14,66],[3,28,0,0,31,70,0,0,0,0,12,51,0,0,63,61] >;
C24.13Q8 in GAP, Magma, Sage, TeX
C_{24}._{13}Q_8 % in TeX
G:=Group("C24.13Q8"); // GroupNames label
G:=SmallGroup(192,242);
// by ID
G=gap.SmallGroup(192,242);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=a^12*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^12*b^-1>;
// generators/relations