metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊12Q8, C8⋊9Dic6, C42.251D6, C12.14M4(2), C6.6(C4×Q8), (C4×C8).16S3, C3⋊1(C8⋊4Q8), (C4×C24).28C2, C6.2(C8○D4), (C2×C8).280D6, C4⋊Dic3.6C4, C12.80(C2×Q8), C12⋊C8.4C2, C24⋊C4.5C2, C2.8(C4×Dic6), C4.6(C8⋊S3), C2.6(C8○D12), Dic3⋊C8.1C2, Dic3⋊C4.1C4, (C4×Dic6).2C2, (C2×Dic6).6C4, C4.45(C2×Dic6), C6.1(C2×M4(2)), C12.239(C4○D4), C4.123(C4○D12), (C4×C12).322C22, (C2×C12).801C23, (C2×C24).339C22, (C4×Dic3).178C22, C2.6(C2×C8⋊S3), C22.94(S3×C2×C4), (C2×C4).102(C4×S3), (C2×C12).219(C2×C4), (C2×C3⋊C8).186C22, (C2×C6).56(C22×C4), (C2×C4).743(C22×S3), (C2×Dic3).10(C2×C4), SmallGroup(192,238)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊12Q8
G = < a,b,c | a24=b4=1, c2=b2, ab=ba, cac-1=a5, cbc-1=b-1 >
Subgroups: 184 in 94 conjugacy classes, 55 normal (33 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, Dic3, C12, 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, C4⋊C8, C4×Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C8⋊4Q8, C12⋊C8, Dic3⋊C8, C24⋊C4, C4×C24, C4×Dic6, C24⋊12Q8
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, C8⋊S3, C2×Dic6, S3×C2×C4, C4○D12, C8⋊4Q8, C4×Dic6, C2×C8⋊S3, C8○D12, C24⋊12Q8
►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)
(1 192 25 61)(2 169 26 62)(3 170 27 63)(4 171 28 64)(5 172 29 65)(6 173 30 66)(7 174 31 67)(8 175 32 68)(9 176 33 69)(10 177 34 70)(11 178 35 71)(12 179 36 72)(13 180 37 49)(14 181 38 50)(15 182 39 51)(16 183 40 52)(17 184 41 53)(18 185 42 54)(19 186 43 55)(20 187 44 56)(21 188 45 57)(22 189 46 58)(23 190 47 59)(24 191 48 60)(73 109 150 141)(74 110 151 142)(75 111 152 143)(76 112 153 144)(77 113 154 121)(78 114 155 122)(79 115 156 123)(80 116 157 124)(81 117 158 125)(82 118 159 126)(83 119 160 127)(84 120 161 128)(85 97 162 129)(86 98 163 130)(87 99 164 131)(88 100 165 132)(89 101 166 133)(90 102 167 134)(91 103 168 135)(92 104 145 136)(93 105 146 137)(94 106 147 138)(95 107 148 139)(96 108 149 140)
(1 117 25 125)(2 98 26 130)(3 103 27 135)(4 108 28 140)(5 113 29 121)(6 118 30 126)(7 99 31 131)(8 104 32 136)(9 109 33 141)(10 114 34 122)(11 119 35 127)(12 100 36 132)(13 105 37 137)(14 110 38 142)(15 115 39 123)(16 120 40 128)(17 101 41 133)(18 106 42 138)(19 111 43 143)(20 116 44 124)(21 97 45 129)(22 102 46 134)(23 107 47 139)(24 112 48 144)(49 146 180 93)(50 151 181 74)(51 156 182 79)(52 161 183 84)(53 166 184 89)(54 147 185 94)(55 152 186 75)(56 157 187 80)(57 162 188 85)(58 167 189 90)(59 148 190 95)(60 153 191 76)(61 158 192 81)(62 163 169 86)(63 168 170 91)(64 149 171 96)(65 154 172 77)(66 159 173 82)(67 164 174 87)(68 145 175 92)(69 150 176 73)(70 155 177 78)(71 160 178 83)(72 165 179 88)
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), (1,192,25,61)(2,169,26,62)(3,170,27,63)(4,171,28,64)(5,172,29,65)(6,173,30,66)(7,174,31,67)(8,175,32,68)(9,176,33,69)(10,177,34,70)(11,178,35,71)(12,179,36,72)(13,180,37,49)(14,181,38,50)(15,182,39,51)(16,183,40,52)(17,184,41,53)(18,185,42,54)(19,186,43,55)(20,187,44,56)(21,188,45,57)(22,189,46,58)(23,190,47,59)(24,191,48,60)(73,109,150,141)(74,110,151,142)(75,111,152,143)(76,112,153,144)(77,113,154,121)(78,114,155,122)(79,115,156,123)(80,116,157,124)(81,117,158,125)(82,118,159,126)(83,119,160,127)(84,120,161,128)(85,97,162,129)(86,98,163,130)(87,99,164,131)(88,100,165,132)(89,101,166,133)(90,102,167,134)(91,103,168,135)(92,104,145,136)(93,105,146,137)(94,106,147,138)(95,107,148,139)(96,108,149,140), (1,117,25,125)(2,98,26,130)(3,103,27,135)(4,108,28,140)(5,113,29,121)(6,118,30,126)(7,99,31,131)(8,104,32,136)(9,109,33,141)(10,114,34,122)(11,119,35,127)(12,100,36,132)(13,105,37,137)(14,110,38,142)(15,115,39,123)(16,120,40,128)(17,101,41,133)(18,106,42,138)(19,111,43,143)(20,116,44,124)(21,97,45,129)(22,102,46,134)(23,107,47,139)(24,112,48,144)(49,146,180,93)(50,151,181,74)(51,156,182,79)(52,161,183,84)(53,166,184,89)(54,147,185,94)(55,152,186,75)(56,157,187,80)(57,162,188,85)(58,167,189,90)(59,148,190,95)(60,153,191,76)(61,158,192,81)(62,163,169,86)(63,168,170,91)(64,149,171,96)(65,154,172,77)(66,159,173,82)(67,164,174,87)(68,145,175,92)(69,150,176,73)(70,155,177,78)(71,160,178,83)(72,165,179,88)>;
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), (1,192,25,61)(2,169,26,62)(3,170,27,63)(4,171,28,64)(5,172,29,65)(6,173,30,66)(7,174,31,67)(8,175,32,68)(9,176,33,69)(10,177,34,70)(11,178,35,71)(12,179,36,72)(13,180,37,49)(14,181,38,50)(15,182,39,51)(16,183,40,52)(17,184,41,53)(18,185,42,54)(19,186,43,55)(20,187,44,56)(21,188,45,57)(22,189,46,58)(23,190,47,59)(24,191,48,60)(73,109,150,141)(74,110,151,142)(75,111,152,143)(76,112,153,144)(77,113,154,121)(78,114,155,122)(79,115,156,123)(80,116,157,124)(81,117,158,125)(82,118,159,126)(83,119,160,127)(84,120,161,128)(85,97,162,129)(86,98,163,130)(87,99,164,131)(88,100,165,132)(89,101,166,133)(90,102,167,134)(91,103,168,135)(92,104,145,136)(93,105,146,137)(94,106,147,138)(95,107,148,139)(96,108,149,140), (1,117,25,125)(2,98,26,130)(3,103,27,135)(4,108,28,140)(5,113,29,121)(6,118,30,126)(7,99,31,131)(8,104,32,136)(9,109,33,141)(10,114,34,122)(11,119,35,127)(12,100,36,132)(13,105,37,137)(14,110,38,142)(15,115,39,123)(16,120,40,128)(17,101,41,133)(18,106,42,138)(19,111,43,143)(20,116,44,124)(21,97,45,129)(22,102,46,134)(23,107,47,139)(24,112,48,144)(49,146,180,93)(50,151,181,74)(51,156,182,79)(52,161,183,84)(53,166,184,89)(54,147,185,94)(55,152,186,75)(56,157,187,80)(57,162,188,85)(58,167,189,90)(59,148,190,95)(60,153,191,76)(61,158,192,81)(62,163,169,86)(63,168,170,91)(64,149,171,96)(65,154,172,77)(66,159,173,82)(67,164,174,87)(68,145,175,92)(69,150,176,73)(70,155,177,78)(71,160,178,83)(72,165,179,88) );
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)], [(1,192,25,61),(2,169,26,62),(3,170,27,63),(4,171,28,64),(5,172,29,65),(6,173,30,66),(7,174,31,67),(8,175,32,68),(9,176,33,69),(10,177,34,70),(11,178,35,71),(12,179,36,72),(13,180,37,49),(14,181,38,50),(15,182,39,51),(16,183,40,52),(17,184,41,53),(18,185,42,54),(19,186,43,55),(20,187,44,56),(21,188,45,57),(22,189,46,58),(23,190,47,59),(24,191,48,60),(73,109,150,141),(74,110,151,142),(75,111,152,143),(76,112,153,144),(77,113,154,121),(78,114,155,122),(79,115,156,123),(80,116,157,124),(81,117,158,125),(82,118,159,126),(83,119,160,127),(84,120,161,128),(85,97,162,129),(86,98,163,130),(87,99,164,131),(88,100,165,132),(89,101,166,133),(90,102,167,134),(91,103,168,135),(92,104,145,136),(93,105,146,137),(94,106,147,138),(95,107,148,139),(96,108,149,140)], [(1,117,25,125),(2,98,26,130),(3,103,27,135),(4,108,28,140),(5,113,29,121),(6,118,30,126),(7,99,31,131),(8,104,32,136),(9,109,33,141),(10,114,34,122),(11,119,35,127),(12,100,36,132),(13,105,37,137),(14,110,38,142),(15,115,39,123),(16,120,40,128),(17,101,41,133),(18,106,42,138),(19,111,43,143),(20,116,44,124),(21,97,45,129),(22,102,46,134),(23,107,47,139),(24,112,48,144),(49,146,180,93),(50,151,181,74),(51,156,182,79),(52,161,183,84),(53,166,184,89),(54,147,185,94),(55,152,186,75),(56,157,187,80),(57,162,188,85),(58,167,189,90),(59,148,190,95),(60,153,191,76),(61,158,192,81),(62,163,169,86),(63,168,170,91),(64,149,171,96),(65,154,172,77),(66,159,173,82),(67,164,174,87),(68,145,175,92),(69,150,176,73),(70,155,177,78),(71,160,178,83),(72,165,179,88)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12L | 24A | ··· | 24P |
| 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 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Q8 | D6 | D6 | M4(2) | C4○D4 | Dic6 | C4×S3 | C8○D4 | C8⋊S3 | C4○D12 | C8○D12 |
| kernel | C24⋊12Q8 | C12⋊C8 | Dic3⋊C8 | C24⋊C4 | C4×C24 | C4×Dic6 | Dic3⋊C4 | C4⋊Dic3 | C2×Dic6 | C4×C8 | C24 | C42 | C2×C8 | C12 | C12 | C8 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C24⋊12Q8 ►in GL4(𝔽73) generated by
| 70 | 3 | 0 | 0 |
| 70 | 67 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 14 | 7 |
| 2 | 13 | 0 | 0 |
| 11 | 71 | 0 | 0 |
| 0 | 0 | 60 | 30 |
| 0 | 0 | 43 | 13 |
G:=sub<GL(4,GF(73))| [70,70,0,0,3,67,0,0,0,0,72,1,0,0,72,0],[1,0,0,0,0,1,0,0,0,0,66,14,0,0,59,7],[2,11,0,0,13,71,0,0,0,0,60,43,0,0,30,13] >;
C24⋊12Q8 in GAP, Magma, Sage, TeX
C_{24}\rtimes_{12}Q_8 % in TeX
G:=Group("C24:12Q8"); // GroupNames label
G:=SmallGroup(192,238);
// by ID
G=gap.SmallGroup(192,238);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,758,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations