metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊9Q8, C8⋊8Dic6, C12.24SD16, C42.252D6, (C4×C8).13S3, C3⋊1(C8⋊3Q8), C6.2(C4⋊Q8), (C4×C24).15C2, (C2×C4).76D12, (C2×C8).315D6, C12.69(C2×Q8), C8⋊Dic3.4C2, C6.1(C2×SD16), C4.3(C24⋊C2), (C2×C12).373D4, C12⋊2Q8.1C2, C4.35(C2×Dic6), C2.6(C12⋊2Q8), C22.85(C2×D12), C4⋊Dic3.1C22, (C2×C12).716C23, (C2×C24).387C22, (C4×C12).302C22, C2.5(C2×C24⋊C2), (C2×C6).99(C2×D4), (C2×C4).659(C22×S3), SmallGroup(192,239)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊9Q8
G = < a,b,c | a24=b4=1, c2=b2, ab=ba, cac-1=a11, cbc-1=b-1 >
Subgroups: 280 in 98 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C4×C8, C4.Q8, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C8⋊3Q8, C8⋊Dic3, C4×C24, C12⋊2Q8, C24⋊9Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, Dic6, D12, C22×S3, C4⋊Q8, C2×SD16, C24⋊C2, C2×Dic6, C2×D12, C8⋊3Q8, C12⋊2Q8, C2×C24⋊C2, C24⋊9Q8
►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)
(1 122 66 178)(2 123 67 179)(3 124 68 180)(4 125 69 181)(5 126 70 182)(6 127 71 183)(7 128 72 184)(8 129 49 185)(9 130 50 186)(10 131 51 187)(11 132 52 188)(12 133 53 189)(13 134 54 190)(14 135 55 191)(15 136 56 192)(16 137 57 169)(17 138 58 170)(18 139 59 171)(19 140 60 172)(20 141 61 173)(21 142 62 174)(22 143 63 175)(23 144 64 176)(24 121 65 177)(25 93 166 101)(26 94 167 102)(27 95 168 103)(28 96 145 104)(29 73 146 105)(30 74 147 106)(31 75 148 107)(32 76 149 108)(33 77 150 109)(34 78 151 110)(35 79 152 111)(36 80 153 112)(37 81 154 113)(38 82 155 114)(39 83 156 115)(40 84 157 116)(41 85 158 117)(42 86 159 118)(43 87 160 119)(44 88 161 120)(45 89 162 97)(46 90 163 98)(47 91 164 99)(48 92 165 100)
(1 90 66 98)(2 77 67 109)(3 88 68 120)(4 75 69 107)(5 86 70 118)(6 73 71 105)(7 84 72 116)(8 95 49 103)(9 82 50 114)(10 93 51 101)(11 80 52 112)(12 91 53 99)(13 78 54 110)(14 89 55 97)(15 76 56 108)(16 87 57 119)(17 74 58 106)(18 85 59 117)(19 96 60 104)(20 83 61 115)(21 94 62 102)(22 81 63 113)(23 92 64 100)(24 79 65 111)(25 187 166 131)(26 174 167 142)(27 185 168 129)(28 172 145 140)(29 183 146 127)(30 170 147 138)(31 181 148 125)(32 192 149 136)(33 179 150 123)(34 190 151 134)(35 177 152 121)(36 188 153 132)(37 175 154 143)(38 186 155 130)(39 173 156 141)(40 184 157 128)(41 171 158 139)(42 182 159 126)(43 169 160 137)(44 180 161 124)(45 191 162 135)(46 178 163 122)(47 189 164 133)(48 176 165 144)
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), (1,122,66,178)(2,123,67,179)(3,124,68,180)(4,125,69,181)(5,126,70,182)(6,127,71,183)(7,128,72,184)(8,129,49,185)(9,130,50,186)(10,131,51,187)(11,132,52,188)(12,133,53,189)(13,134,54,190)(14,135,55,191)(15,136,56,192)(16,137,57,169)(17,138,58,170)(18,139,59,171)(19,140,60,172)(20,141,61,173)(21,142,62,174)(22,143,63,175)(23,144,64,176)(24,121,65,177)(25,93,166,101)(26,94,167,102)(27,95,168,103)(28,96,145,104)(29,73,146,105)(30,74,147,106)(31,75,148,107)(32,76,149,108)(33,77,150,109)(34,78,151,110)(35,79,152,111)(36,80,153,112)(37,81,154,113)(38,82,155,114)(39,83,156,115)(40,84,157,116)(41,85,158,117)(42,86,159,118)(43,87,160,119)(44,88,161,120)(45,89,162,97)(46,90,163,98)(47,91,164,99)(48,92,165,100), (1,90,66,98)(2,77,67,109)(3,88,68,120)(4,75,69,107)(5,86,70,118)(6,73,71,105)(7,84,72,116)(8,95,49,103)(9,82,50,114)(10,93,51,101)(11,80,52,112)(12,91,53,99)(13,78,54,110)(14,89,55,97)(15,76,56,108)(16,87,57,119)(17,74,58,106)(18,85,59,117)(19,96,60,104)(20,83,61,115)(21,94,62,102)(22,81,63,113)(23,92,64,100)(24,79,65,111)(25,187,166,131)(26,174,167,142)(27,185,168,129)(28,172,145,140)(29,183,146,127)(30,170,147,138)(31,181,148,125)(32,192,149,136)(33,179,150,123)(34,190,151,134)(35,177,152,121)(36,188,153,132)(37,175,154,143)(38,186,155,130)(39,173,156,141)(40,184,157,128)(41,171,158,139)(42,182,159,126)(43,169,160,137)(44,180,161,124)(45,191,162,135)(46,178,163,122)(47,189,164,133)(48,176,165,144)>;
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), (1,122,66,178)(2,123,67,179)(3,124,68,180)(4,125,69,181)(5,126,70,182)(6,127,71,183)(7,128,72,184)(8,129,49,185)(9,130,50,186)(10,131,51,187)(11,132,52,188)(12,133,53,189)(13,134,54,190)(14,135,55,191)(15,136,56,192)(16,137,57,169)(17,138,58,170)(18,139,59,171)(19,140,60,172)(20,141,61,173)(21,142,62,174)(22,143,63,175)(23,144,64,176)(24,121,65,177)(25,93,166,101)(26,94,167,102)(27,95,168,103)(28,96,145,104)(29,73,146,105)(30,74,147,106)(31,75,148,107)(32,76,149,108)(33,77,150,109)(34,78,151,110)(35,79,152,111)(36,80,153,112)(37,81,154,113)(38,82,155,114)(39,83,156,115)(40,84,157,116)(41,85,158,117)(42,86,159,118)(43,87,160,119)(44,88,161,120)(45,89,162,97)(46,90,163,98)(47,91,164,99)(48,92,165,100), (1,90,66,98)(2,77,67,109)(3,88,68,120)(4,75,69,107)(5,86,70,118)(6,73,71,105)(7,84,72,116)(8,95,49,103)(9,82,50,114)(10,93,51,101)(11,80,52,112)(12,91,53,99)(13,78,54,110)(14,89,55,97)(15,76,56,108)(16,87,57,119)(17,74,58,106)(18,85,59,117)(19,96,60,104)(20,83,61,115)(21,94,62,102)(22,81,63,113)(23,92,64,100)(24,79,65,111)(25,187,166,131)(26,174,167,142)(27,185,168,129)(28,172,145,140)(29,183,146,127)(30,170,147,138)(31,181,148,125)(32,192,149,136)(33,179,150,123)(34,190,151,134)(35,177,152,121)(36,188,153,132)(37,175,154,143)(38,186,155,130)(39,173,156,141)(40,184,157,128)(41,171,158,139)(42,182,159,126)(43,169,160,137)(44,180,161,124)(45,191,162,135)(46,178,163,122)(47,189,164,133)(48,176,165,144) );
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)], [(1,122,66,178),(2,123,67,179),(3,124,68,180),(4,125,69,181),(5,126,70,182),(6,127,71,183),(7,128,72,184),(8,129,49,185),(9,130,50,186),(10,131,51,187),(11,132,52,188),(12,133,53,189),(13,134,54,190),(14,135,55,191),(15,136,56,192),(16,137,57,169),(17,138,58,170),(18,139,59,171),(19,140,60,172),(20,141,61,173),(21,142,62,174),(22,143,63,175),(23,144,64,176),(24,121,65,177),(25,93,166,101),(26,94,167,102),(27,95,168,103),(28,96,145,104),(29,73,146,105),(30,74,147,106),(31,75,148,107),(32,76,149,108),(33,77,150,109),(34,78,151,110),(35,79,152,111),(36,80,153,112),(37,81,154,113),(38,82,155,114),(39,83,156,115),(40,84,157,116),(41,85,158,117),(42,86,159,118),(43,87,160,119),(44,88,161,120),(45,89,162,97),(46,90,163,98),(47,91,164,99),(48,92,165,100)], [(1,90,66,98),(2,77,67,109),(3,88,68,120),(4,75,69,107),(5,86,70,118),(6,73,71,105),(7,84,72,116),(8,95,49,103),(9,82,50,114),(10,93,51,101),(11,80,52,112),(12,91,53,99),(13,78,54,110),(14,89,55,97),(15,76,56,108),(16,87,57,119),(17,74,58,106),(18,85,59,117),(19,96,60,104),(20,83,61,115),(21,94,62,102),(22,81,63,113),(23,92,64,100),(24,79,65,111),(25,187,166,131),(26,174,167,142),(27,185,168,129),(28,172,145,140),(29,183,146,127),(30,170,147,138),(31,181,148,125),(32,192,149,136),(33,179,150,123),(34,190,151,134),(35,177,152,121),(36,188,153,132),(37,175,154,143),(38,186,155,130),(39,173,156,141),(40,184,157,128),(41,171,158,139),(42,182,159,126),(43,169,160,137),(44,180,161,124),(45,191,162,135),(46,178,163,122),(47,189,164,133),(48,176,165,144)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | SD16 | Dic6 | D12 | C24⋊C2 |
| kernel | C24⋊9Q8 | C8⋊Dic3 | C4×C24 | C12⋊2Q8 | C4×C8 | C24 | C2×C12 | C42 | C2×C8 | C12 | C8 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 16 |
Matrix representation of C24⋊9Q8 ►in GL4(𝔽73) generated by
| 25 | 36 | 0 | 0 |
| 37 | 62 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 51 | 10 | 0 | 0 |
| 32 | 22 | 0 | 0 |
| 0 | 0 | 30 | 11 |
| 0 | 0 | 11 | 43 |
G:=sub<GL(4,GF(73))| [25,37,0,0,36,62,0,0,0,0,0,1,0,0,72,0],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[51,32,0,0,10,22,0,0,0,0,30,11,0,0,11,43] >;
C24⋊9Q8 in GAP, Magma, Sage, TeX
C_{24}\rtimes_9Q_8 % in TeX
G:=Group("C24:9Q8"); // GroupNames label
G:=SmallGroup(192,239);
// by ID
G=gap.SmallGroup(192,239);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^11,c*b*c^-1=b^-1>;
// generators/relations