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## G = C92.C4order 368 = 24·23

### 1st non-split extension by C92 of C4 acting via C4/C2=C2

Aliases: C92.1C4, C4.Dic23, C4.15D46, C232M4(2), C22.Dic23, C92.15C22, C23⋊C85C2, (C2×C46).3C4, (C2×C92).5C2, C46.7(C2×C4), (C2×C4).2D23, C2.3(C2×Dic23), SmallGroup(368,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C46 — C92.C4
 Chief series C1 — C23 — C46 — C92 — C23⋊C8 — C92.C4
 Lower central C23 — C46 — C92.C4
 Upper central C1 — C4 — C2×C4

Generators and relations for C92.C4
G = < a,b | a92=1, b4=a46, bab-1=a-1 >

Smallest permutation representation of C92.C4
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 95 70 118 47 141 24 164)(2 94 71 117 48 140 25 163)(3 93 72 116 49 139 26 162)(4 184 73 115 50 138 27 161)(5 183 74 114 51 137 28 160)(6 182 75 113 52 136 29 159)(7 181 76 112 53 135 30 158)(8 180 77 111 54 134 31 157)(9 179 78 110 55 133 32 156)(10 178 79 109 56 132 33 155)(11 177 80 108 57 131 34 154)(12 176 81 107 58 130 35 153)(13 175 82 106 59 129 36 152)(14 174 83 105 60 128 37 151)(15 173 84 104 61 127 38 150)(16 172 85 103 62 126 39 149)(17 171 86 102 63 125 40 148)(18 170 87 101 64 124 41 147)(19 169 88 100 65 123 42 146)(20 168 89 99 66 122 43 145)(21 167 90 98 67 121 44 144)(22 166 91 97 68 120 45 143)(23 165 92 96 69 119 46 142)```

`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), (1,95,70,118,47,141,24,164)(2,94,71,117,48,140,25,163)(3,93,72,116,49,139,26,162)(4,184,73,115,50,138,27,161)(5,183,74,114,51,137,28,160)(6,182,75,113,52,136,29,159)(7,181,76,112,53,135,30,158)(8,180,77,111,54,134,31,157)(9,179,78,110,55,133,32,156)(10,178,79,109,56,132,33,155)(11,177,80,108,57,131,34,154)(12,176,81,107,58,130,35,153)(13,175,82,106,59,129,36,152)(14,174,83,105,60,128,37,151)(15,173,84,104,61,127,38,150)(16,172,85,103,62,126,39,149)(17,171,86,102,63,125,40,148)(18,170,87,101,64,124,41,147)(19,169,88,100,65,123,42,146)(20,168,89,99,66,122,43,145)(21,167,90,98,67,121,44,144)(22,166,91,97,68,120,45,143)(23,165,92,96,69,119,46,142)>;`

`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), (1,95,70,118,47,141,24,164)(2,94,71,117,48,140,25,163)(3,93,72,116,49,139,26,162)(4,184,73,115,50,138,27,161)(5,183,74,114,51,137,28,160)(6,182,75,113,52,136,29,159)(7,181,76,112,53,135,30,158)(8,180,77,111,54,134,31,157)(9,179,78,110,55,133,32,156)(10,178,79,109,56,132,33,155)(11,177,80,108,57,131,34,154)(12,176,81,107,58,130,35,153)(13,175,82,106,59,129,36,152)(14,174,83,105,60,128,37,151)(15,173,84,104,61,127,38,150)(16,172,85,103,62,126,39,149)(17,171,86,102,63,125,40,148)(18,170,87,101,64,124,41,147)(19,169,88,100,65,123,42,146)(20,168,89,99,66,122,43,145)(21,167,90,98,67,121,44,144)(22,166,91,97,68,120,45,143)(23,165,92,96,69,119,46,142) );`

`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)], [(1,95,70,118,47,141,24,164),(2,94,71,117,48,140,25,163),(3,93,72,116,49,139,26,162),(4,184,73,115,50,138,27,161),(5,183,74,114,51,137,28,160),(6,182,75,113,52,136,29,159),(7,181,76,112,53,135,30,158),(8,180,77,111,54,134,31,157),(9,179,78,110,55,133,32,156),(10,178,79,109,56,132,33,155),(11,177,80,108,57,131,34,154),(12,176,81,107,58,130,35,153),(13,175,82,106,59,129,36,152),(14,174,83,105,60,128,37,151),(15,173,84,104,61,127,38,150),(16,172,85,103,62,126,39,149),(17,171,86,102,63,125,40,148),(18,170,87,101,64,124,41,147),(19,169,88,100,65,123,42,146),(20,168,89,99,66,122,43,145),(21,167,90,98,67,121,44,144),(22,166,91,97,68,120,45,143),(23,165,92,96,69,119,46,142)]])`

98 conjugacy classes

 class 1 2A 2B 4A 4B 4C 8A 8B 8C 8D 23A ··· 23K 46A ··· 46AG 92A ··· 92AR order 1 2 2 4 4 4 8 8 8 8 23 ··· 23 46 ··· 46 92 ··· 92 size 1 1 2 1 1 2 46 46 46 46 2 ··· 2 2 ··· 2 2 ··· 2

98 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 M4(2) D23 Dic23 D46 Dic23 C92.C4 kernel C92.C4 C23⋊C8 C2×C92 C92 C2×C46 C23 C2×C4 C4 C4 C22 C1 # reps 1 2 1 2 2 2 11 11 11 11 44

Matrix representation of C92.C4 in GL2(𝔽1289) generated by

 62 0 1052 894
,
 1094 1223 647 195
`G:=sub<GL(2,GF(1289))| [62,1052,0,894],[1094,647,1223,195] >;`

C92.C4 in GAP, Magma, Sage, TeX

`C_{92}.C_4`
`% in TeX`

`G:=Group("C92.C4");`
`// GroupNames label`

`G:=SmallGroup(368,9);`
`// by ID`

`G=gap.SmallGroup(368,9);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-23,20,101,42,8804]);`
`// Polycyclic`

`G:=Group<a,b|a^92=1,b^4=a^46,b*a*b^-1=a^-1>;`
`// generators/relations`

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