metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24.4C8, C6.4M5(2), C12.14C42, C42.4Dic3, C3⋊C16⋊7C4, C8.2(C3⋊C8), C6.4(C4×C8), C8.39(C4×S3), (C4×C8).15S3, (C2×C12).3C8, C3⋊1(C16⋊5C4), (C2×C24).24C4, C24.60(C2×C4), C12.41(C2×C8), (C4×C24).25C2, (C4×C12).18C4, (C2×C8).330D6, C4.14(C4×Dic3), (C2×C8).13Dic3, C2.1(C12.C8), (C2×C24).414C22, C2.4(C4×C3⋊C8), C4.13(C2×C3⋊C8), (C2×C3⋊C16).7C2, (C2×C4).2(C3⋊C8), C22.8(C2×C3⋊C8), (C2×C6).26(C2×C8), (C2×C12).310(C2×C4), (C2×C4).91(C2×Dic3), SmallGroup(192,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.C8
G = < a,b | a24=1, b8=a12, bab-1=a5 >
Smallest permutation representation of C24.C8
►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)
(1 159 64 73 114 144 43 176 13 147 52 85 102 132 31 188)(2 164 65 78 115 125 44 181 14 152 53 90 103 137 32 169)(3 145 66 83 116 130 45 186 15 157 54 95 104 142 33 174)(4 150 67 88 117 135 46 191 16 162 55 76 105 123 34 179)(5 155 68 93 118 140 47 172 17 167 56 81 106 128 35 184)(6 160 69 74 119 121 48 177 18 148 57 86 107 133 36 189)(7 165 70 79 120 126 25 182 19 153 58 91 108 138 37 170)(8 146 71 84 97 131 26 187 20 158 59 96 109 143 38 175)(9 151 72 89 98 136 27 192 21 163 60 77 110 124 39 180)(10 156 49 94 99 141 28 173 22 168 61 82 111 129 40 185)(11 161 50 75 100 122 29 178 23 149 62 87 112 134 41 190)(12 166 51 80 101 127 30 183 24 154 63 92 113 139 42 171)
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), (1,159,64,73,114,144,43,176,13,147,52,85,102,132,31,188)(2,164,65,78,115,125,44,181,14,152,53,90,103,137,32,169)(3,145,66,83,116,130,45,186,15,157,54,95,104,142,33,174)(4,150,67,88,117,135,46,191,16,162,55,76,105,123,34,179)(5,155,68,93,118,140,47,172,17,167,56,81,106,128,35,184)(6,160,69,74,119,121,48,177,18,148,57,86,107,133,36,189)(7,165,70,79,120,126,25,182,19,153,58,91,108,138,37,170)(8,146,71,84,97,131,26,187,20,158,59,96,109,143,38,175)(9,151,72,89,98,136,27,192,21,163,60,77,110,124,39,180)(10,156,49,94,99,141,28,173,22,168,61,82,111,129,40,185)(11,161,50,75,100,122,29,178,23,149,62,87,112,134,41,190)(12,166,51,80,101,127,30,183,24,154,63,92,113,139,42,171)>;
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), (1,159,64,73,114,144,43,176,13,147,52,85,102,132,31,188)(2,164,65,78,115,125,44,181,14,152,53,90,103,137,32,169)(3,145,66,83,116,130,45,186,15,157,54,95,104,142,33,174)(4,150,67,88,117,135,46,191,16,162,55,76,105,123,34,179)(5,155,68,93,118,140,47,172,17,167,56,81,106,128,35,184)(6,160,69,74,119,121,48,177,18,148,57,86,107,133,36,189)(7,165,70,79,120,126,25,182,19,153,58,91,108,138,37,170)(8,146,71,84,97,131,26,187,20,158,59,96,109,143,38,175)(9,151,72,89,98,136,27,192,21,163,60,77,110,124,39,180)(10,156,49,94,99,141,28,173,22,168,61,82,111,129,40,185)(11,161,50,75,100,122,29,178,23,149,62,87,112,134,41,190)(12,166,51,80,101,127,30,183,24,154,63,92,113,139,42,171) );
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)], [(1,159,64,73,114,144,43,176,13,147,52,85,102,132,31,188),(2,164,65,78,115,125,44,181,14,152,53,90,103,137,32,169),(3,145,66,83,116,130,45,186,15,157,54,95,104,142,33,174),(4,150,67,88,117,135,46,191,16,162,55,76,105,123,34,179),(5,155,68,93,118,140,47,172,17,167,56,81,106,128,35,184),(6,160,69,74,119,121,48,177,18,148,57,86,107,133,36,189),(7,165,70,79,120,126,25,182,19,153,58,91,108,138,37,170),(8,146,71,84,97,131,26,187,20,158,59,96,109,143,38,175),(9,151,72,89,98,136,27,192,21,163,60,77,110,124,39,180),(10,156,49,94,99,141,28,173,22,168,61,82,111,129,40,185),(11,161,50,75,100,122,29,178,23,149,62,87,112,134,41,190),(12,166,51,80,101,127,30,183,24,154,63,92,113,139,42,171)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12L | 16A | ··· | 16P | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | S3 | Dic3 | Dic3 | D6 | C3⋊C8 | C4×S3 | C3⋊C8 | M5(2) | C12.C8 |
| kernel | C24.C8 | C2×C3⋊C16 | C4×C24 | C3⋊C16 | C4×C12 | C2×C24 | C24 | C2×C12 | C4×C8 | C42 | C2×C8 | C2×C8 | C8 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 16 |
Matrix representation of C24.C8 ►in GL5(𝔽97)
| 75 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 47 | 44 |
| 0 | 0 | 0 | 0 | 50 |
| 22 | 0 | 0 | 0 | 0 |
| 0 | 88 | 7 | 0 | 0 |
| 0 | 16 | 9 | 0 | 0 |
| 0 | 0 | 0 | 53 | 80 |
| 0 | 0 | 0 | 94 | 44 |
G:=sub<GL(5,GF(97))| [75,0,0,0,0,0,1,96,0,0,0,1,0,0,0,0,0,0,47,0,0,0,0,44,50],[22,0,0,0,0,0,88,16,0,0,0,7,9,0,0,0,0,0,53,94,0,0,0,80,44] >;
C24.C8 in GAP, Magma, Sage, TeX
C_{24}.C_8 % in TeX
G:=Group("C24.C8"); // GroupNames label
G:=SmallGroup(192,20);
// by ID
G=gap.SmallGroup(192,20);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,477,64,100,102,6278]);
// Polycyclic
G:=Group<a,b|a^24=1,b^8=a^12,b*a*b^-1=a^5>;
// generators/relations
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