metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C23⋊C8, C46.C4, C92.2C2, C4.2D23, C2.Dic23, SmallGroup(184,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C23⋊C8 |
Generators and relations for C23⋊C8
G = < a,b | a23=b8=1, bab-1=a-1 >
Smallest permutation representation of C23⋊C8
►Regular action on 184 points
Generators in S184
(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)
(1 162 70 116 29 139 52 93)(2 184 71 138 30 161 53 115)(3 183 72 137 31 160 54 114)(4 182 73 136 32 159 55 113)(5 181 74 135 33 158 56 112)(6 180 75 134 34 157 57 111)(7 179 76 133 35 156 58 110)(8 178 77 132 36 155 59 109)(9 177 78 131 37 154 60 108)(10 176 79 130 38 153 61 107)(11 175 80 129 39 152 62 106)(12 174 81 128 40 151 63 105)(13 173 82 127 41 150 64 104)(14 172 83 126 42 149 65 103)(15 171 84 125 43 148 66 102)(16 170 85 124 44 147 67 101)(17 169 86 123 45 146 68 100)(18 168 87 122 46 145 69 99)(19 167 88 121 24 144 47 98)(20 166 89 120 25 143 48 97)(21 165 90 119 26 142 49 96)(22 164 91 118 27 141 50 95)(23 163 92 117 28 140 51 94)
G:=sub<Sym(184)| (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), (1,162,70,116,29,139,52,93)(2,184,71,138,30,161,53,115)(3,183,72,137,31,160,54,114)(4,182,73,136,32,159,55,113)(5,181,74,135,33,158,56,112)(6,180,75,134,34,157,57,111)(7,179,76,133,35,156,58,110)(8,178,77,132,36,155,59,109)(9,177,78,131,37,154,60,108)(10,176,79,130,38,153,61,107)(11,175,80,129,39,152,62,106)(12,174,81,128,40,151,63,105)(13,173,82,127,41,150,64,104)(14,172,83,126,42,149,65,103)(15,171,84,125,43,148,66,102)(16,170,85,124,44,147,67,101)(17,169,86,123,45,146,68,100)(18,168,87,122,46,145,69,99)(19,167,88,121,24,144,47,98)(20,166,89,120,25,143,48,97)(21,165,90,119,26,142,49,96)(22,164,91,118,27,141,50,95)(23,163,92,117,28,140,51,94)>;
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), (1,162,70,116,29,139,52,93)(2,184,71,138,30,161,53,115)(3,183,72,137,31,160,54,114)(4,182,73,136,32,159,55,113)(5,181,74,135,33,158,56,112)(6,180,75,134,34,157,57,111)(7,179,76,133,35,156,58,110)(8,178,77,132,36,155,59,109)(9,177,78,131,37,154,60,108)(10,176,79,130,38,153,61,107)(11,175,80,129,39,152,62,106)(12,174,81,128,40,151,63,105)(13,173,82,127,41,150,64,104)(14,172,83,126,42,149,65,103)(15,171,84,125,43,148,66,102)(16,170,85,124,44,147,67,101)(17,169,86,123,45,146,68,100)(18,168,87,122,46,145,69,99)(19,167,88,121,24,144,47,98)(20,166,89,120,25,143,48,97)(21,165,90,119,26,142,49,96)(22,164,91,118,27,141,50,95)(23,163,92,117,28,140,51,94) );
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)], [(1,162,70,116,29,139,52,93),(2,184,71,138,30,161,53,115),(3,183,72,137,31,160,54,114),(4,182,73,136,32,159,55,113),(5,181,74,135,33,158,56,112),(6,180,75,134,34,157,57,111),(7,179,76,133,35,156,58,110),(8,178,77,132,36,155,59,109),(9,177,78,131,37,154,60,108),(10,176,79,130,38,153,61,107),(11,175,80,129,39,152,62,106),(12,174,81,128,40,151,63,105),(13,173,82,127,41,150,64,104),(14,172,83,126,42,149,65,103),(15,171,84,125,43,148,66,102),(16,170,85,124,44,147,67,101),(17,169,86,123,45,146,68,100),(18,168,87,122,46,145,69,99),(19,167,88,121,24,144,47,98),(20,166,89,120,25,143,48,97),(21,165,90,119,26,142,49,96),(22,164,91,118,27,141,50,95),(23,163,92,117,28,140,51,94)]])
C23⋊C8 is a maximal subgroup of
C8×D23 C8⋊D23 C92.C4 D4⋊D23 D4.D23 Q8⋊D23 C23⋊Q16
C23⋊C8 is a maximal quotient of C23⋊C16
52 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92V |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 1 | 1 | 23 | 23 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D23 | Dic23 | C23⋊C8 |
| kernel | C23⋊C8 | C92 | C46 | C23 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 11 | 11 | 22 |
Matrix representation of C23⋊C8 ►in GL3(𝔽1289) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1288 | 408 |
| 887 | 0 | 0 |
| 0 | 290 | 1252 |
| 0 | 984 | 999 |
G:=sub<GL(3,GF(1289))| [1,0,0,0,0,1288,0,1,408],[887,0,0,0,290,984,0,1252,999] >;
C23⋊C8 in GAP, Magma, Sage, TeX
C_{23}\rtimes C_8 % in TeX
G:=Group("C23:C8"); // GroupNames label
G:=SmallGroup(184,1);
// by ID
G=gap.SmallGroup(184,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-23,8,21,2819]);
// Polycyclic
G:=Group<a,b|a^23=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export