metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊5Q8, C8⋊5Dic6, Dic3.3SD16, C3⋊C8⋊5Q8, C4⋊C4.32D6, C3⋊2(C8⋊3Q8), C12⋊Q8.5C2, C4.20(S3×Q8), C4.Q8.5S3, (C2×C8).256D6, C6.13(C4⋊Q8), C2.8(C12⋊Q8), C12.10(C2×Q8), (C8×Dic3).6C2, C8⋊Dic3.13C2, C2.21(S3×SD16), C6.36(C2×SD16), C4.20(C2×Dic6), (C2×Dic3).96D4, C22.210(S3×D4), C12.Q8.5C2, (C2×C24).157C22, (C2×C12).271C23, C4⋊Dic3.103C22, (C4×Dic3).229C22, (C3×C4.Q8).4C2, (C2×C6).276(C2×D4), (C3×C4⋊C4).64C22, (C2×C3⋊C8).224C22, (C2×C4).374(C22×S3), SmallGroup(192,414)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊5Q8
G = < a,b,c | a24=b4=1, c2=b2, bab-1=a19, cac-1=a17, cbc-1=b-1 >
Subgroups: 272 in 98 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, 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, C4.Q8, C4.Q8, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C8⋊3Q8, C12.Q8, C8×Dic3, C8⋊Dic3, C3×C4.Q8, C12⋊Q8, C24⋊5Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, Dic6, C22×S3, C4⋊Q8, C2×SD16, C2×Dic6, S3×D4, S3×Q8, C8⋊3Q8, C12⋊Q8, S3×SD16, C24⋊5Q8
►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)
(1 145 132 170)(2 164 133 189)(3 159 134 184)(4 154 135 179)(5 149 136 174)(6 168 137 169)(7 163 138 188)(8 158 139 183)(9 153 140 178)(10 148 141 173)(11 167 142 192)(12 162 143 187)(13 157 144 182)(14 152 121 177)(15 147 122 172)(16 166 123 191)(17 161 124 186)(18 156 125 181)(19 151 126 176)(20 146 127 171)(21 165 128 190)(22 160 129 185)(23 155 130 180)(24 150 131 175)(25 63 74 117)(26 58 75 112)(27 53 76 107)(28 72 77 102)(29 67 78 97)(30 62 79 116)(31 57 80 111)(32 52 81 106)(33 71 82 101)(34 66 83 120)(35 61 84 115)(36 56 85 110)(37 51 86 105)(38 70 87 100)(39 65 88 119)(40 60 89 114)(41 55 90 109)(42 50 91 104)(43 69 92 99)(44 64 93 118)(45 59 94 113)(46 54 95 108)(47 49 96 103)(48 68 73 98)
(1 81 132 32)(2 74 133 25)(3 91 134 42)(4 84 135 35)(5 77 136 28)(6 94 137 45)(7 87 138 38)(8 80 139 31)(9 73 140 48)(10 90 141 41)(11 83 142 34)(12 76 143 27)(13 93 144 44)(14 86 121 37)(15 79 122 30)(16 96 123 47)(17 89 124 40)(18 82 125 33)(19 75 126 26)(20 92 127 43)(21 85 128 36)(22 78 129 29)(23 95 130 46)(24 88 131 39)(49 191 103 166)(50 184 104 159)(51 177 105 152)(52 170 106 145)(53 187 107 162)(54 180 108 155)(55 173 109 148)(56 190 110 165)(57 183 111 158)(58 176 112 151)(59 169 113 168)(60 186 114 161)(61 179 115 154)(62 172 116 147)(63 189 117 164)(64 182 118 157)(65 175 119 150)(66 192 120 167)(67 185 97 160)(68 178 98 153)(69 171 99 146)(70 188 100 163)(71 181 101 156)(72 174 102 149)
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), (1,145,132,170)(2,164,133,189)(3,159,134,184)(4,154,135,179)(5,149,136,174)(6,168,137,169)(7,163,138,188)(8,158,139,183)(9,153,140,178)(10,148,141,173)(11,167,142,192)(12,162,143,187)(13,157,144,182)(14,152,121,177)(15,147,122,172)(16,166,123,191)(17,161,124,186)(18,156,125,181)(19,151,126,176)(20,146,127,171)(21,165,128,190)(22,160,129,185)(23,155,130,180)(24,150,131,175)(25,63,74,117)(26,58,75,112)(27,53,76,107)(28,72,77,102)(29,67,78,97)(30,62,79,116)(31,57,80,111)(32,52,81,106)(33,71,82,101)(34,66,83,120)(35,61,84,115)(36,56,85,110)(37,51,86,105)(38,70,87,100)(39,65,88,119)(40,60,89,114)(41,55,90,109)(42,50,91,104)(43,69,92,99)(44,64,93,118)(45,59,94,113)(46,54,95,108)(47,49,96,103)(48,68,73,98), (1,81,132,32)(2,74,133,25)(3,91,134,42)(4,84,135,35)(5,77,136,28)(6,94,137,45)(7,87,138,38)(8,80,139,31)(9,73,140,48)(10,90,141,41)(11,83,142,34)(12,76,143,27)(13,93,144,44)(14,86,121,37)(15,79,122,30)(16,96,123,47)(17,89,124,40)(18,82,125,33)(19,75,126,26)(20,92,127,43)(21,85,128,36)(22,78,129,29)(23,95,130,46)(24,88,131,39)(49,191,103,166)(50,184,104,159)(51,177,105,152)(52,170,106,145)(53,187,107,162)(54,180,108,155)(55,173,109,148)(56,190,110,165)(57,183,111,158)(58,176,112,151)(59,169,113,168)(60,186,114,161)(61,179,115,154)(62,172,116,147)(63,189,117,164)(64,182,118,157)(65,175,119,150)(66,192,120,167)(67,185,97,160)(68,178,98,153)(69,171,99,146)(70,188,100,163)(71,181,101,156)(72,174,102,149)>;
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), (1,145,132,170)(2,164,133,189)(3,159,134,184)(4,154,135,179)(5,149,136,174)(6,168,137,169)(7,163,138,188)(8,158,139,183)(9,153,140,178)(10,148,141,173)(11,167,142,192)(12,162,143,187)(13,157,144,182)(14,152,121,177)(15,147,122,172)(16,166,123,191)(17,161,124,186)(18,156,125,181)(19,151,126,176)(20,146,127,171)(21,165,128,190)(22,160,129,185)(23,155,130,180)(24,150,131,175)(25,63,74,117)(26,58,75,112)(27,53,76,107)(28,72,77,102)(29,67,78,97)(30,62,79,116)(31,57,80,111)(32,52,81,106)(33,71,82,101)(34,66,83,120)(35,61,84,115)(36,56,85,110)(37,51,86,105)(38,70,87,100)(39,65,88,119)(40,60,89,114)(41,55,90,109)(42,50,91,104)(43,69,92,99)(44,64,93,118)(45,59,94,113)(46,54,95,108)(47,49,96,103)(48,68,73,98), (1,81,132,32)(2,74,133,25)(3,91,134,42)(4,84,135,35)(5,77,136,28)(6,94,137,45)(7,87,138,38)(8,80,139,31)(9,73,140,48)(10,90,141,41)(11,83,142,34)(12,76,143,27)(13,93,144,44)(14,86,121,37)(15,79,122,30)(16,96,123,47)(17,89,124,40)(18,82,125,33)(19,75,126,26)(20,92,127,43)(21,85,128,36)(22,78,129,29)(23,95,130,46)(24,88,131,39)(49,191,103,166)(50,184,104,159)(51,177,105,152)(52,170,106,145)(53,187,107,162)(54,180,108,155)(55,173,109,148)(56,190,110,165)(57,183,111,158)(58,176,112,151)(59,169,113,168)(60,186,114,161)(61,179,115,154)(62,172,116,147)(63,189,117,164)(64,182,118,157)(65,175,119,150)(66,192,120,167)(67,185,97,160)(68,178,98,153)(69,171,99,146)(70,188,100,163)(71,181,101,156)(72,174,102,149) );
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)], [(1,145,132,170),(2,164,133,189),(3,159,134,184),(4,154,135,179),(5,149,136,174),(6,168,137,169),(7,163,138,188),(8,158,139,183),(9,153,140,178),(10,148,141,173),(11,167,142,192),(12,162,143,187),(13,157,144,182),(14,152,121,177),(15,147,122,172),(16,166,123,191),(17,161,124,186),(18,156,125,181),(19,151,126,176),(20,146,127,171),(21,165,128,190),(22,160,129,185),(23,155,130,180),(24,150,131,175),(25,63,74,117),(26,58,75,112),(27,53,76,107),(28,72,77,102),(29,67,78,97),(30,62,79,116),(31,57,80,111),(32,52,81,106),(33,71,82,101),(34,66,83,120),(35,61,84,115),(36,56,85,110),(37,51,86,105),(38,70,87,100),(39,65,88,119),(40,60,89,114),(41,55,90,109),(42,50,91,104),(43,69,92,99),(44,64,93,118),(45,59,94,113),(46,54,95,108),(47,49,96,103),(48,68,73,98)], [(1,81,132,32),(2,74,133,25),(3,91,134,42),(4,84,135,35),(5,77,136,28),(6,94,137,45),(7,87,138,38),(8,80,139,31),(9,73,140,48),(10,90,141,41),(11,83,142,34),(12,76,143,27),(13,93,144,44),(14,86,121,37),(15,79,122,30),(16,96,123,47),(17,89,124,40),(18,82,125,33),(19,75,126,26),(20,92,127,43),(21,85,128,36),(22,78,129,29),(23,95,130,46),(24,88,131,39),(49,191,103,166),(50,184,104,159),(51,177,105,152),(52,170,106,145),(53,187,107,162),(54,180,108,155),(55,173,109,148),(56,190,110,165),(57,183,111,158),(58,176,112,151),(59,169,113,168),(60,186,114,161),(61,179,115,154),(62,172,116,147),(63,189,117,164),(64,182,118,157),(65,175,119,150),(66,192,120,167),(67,185,97,160),(68,178,98,153),(69,171,99,146),(70,188,100,163),(71,181,101,156),(72,174,102,149)]])
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 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | Q8 | D4 | D6 | D6 | SD16 | Dic6 | S3×Q8 | S3×D4 | S3×SD16 |
| kernel | C24⋊5Q8 | C12.Q8 | C8×Dic3 | C8⋊Dic3 | C3×C4.Q8 | C12⋊Q8 | C4.Q8 | C3⋊C8 | C24 | C2×Dic3 | C4⋊C4 | C2×C8 | Dic3 | C8 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 8 | 4 | 1 | 1 | 4 |
Matrix representation of C24⋊5Q8 ►in GL4(𝔽73) generated by
| 6 | 67 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 57 | 20 | 0 | 0 |
| 20 | 16 | 0 | 0 |
| 0 | 0 | 7 | 59 |
| 0 | 0 | 14 | 66 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 26 | 8 |
| 0 | 0 | 34 | 47 |
G:=sub<GL(4,GF(73))| [6,6,0,0,67,6,0,0,0,0,0,1,0,0,72,72],[57,20,0,0,20,16,0,0,0,0,7,14,0,0,59,66],[0,72,0,0,1,0,0,0,0,0,26,34,0,0,8,47] >;
C24⋊5Q8 in GAP, Magma, Sage, TeX
C_{24}\rtimes_5Q_8 % in TeX
G:=Group("C24:5Q8"); // GroupNames label
G:=SmallGroup(192,414);
// by ID
G=gap.SmallGroup(192,414);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,254,555,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=b^2,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=b^-1>;
// generators/relations