Copied to
clipboard

## G = C12.26Q16order 192 = 26·3

### 5th non-split extension by C12 of Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.26Q16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×C12 — C12⋊C8 — C12.26Q16
 Lower central C3 — C6 — C12 — C12.26Q16
 Upper central C1 — C2×C4 — C42 — C4×Q8

Generators and relations for C12.26Q16
G = < a,b,c | a12=b8=1, c2=a6b4, bab-1=a5, ac=ca, cbc-1=a9b-1 >

Subgroups: 136 in 70 conjugacy classes, 39 normal (35 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C2×C12, C2×C12, C3×Q8, C3×Q8, C4×C8, C4⋊C8, C4×Q8, C2×C3⋊C8, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C6×Q8, Q8⋊C8, C4×C3⋊C8, C12⋊C8, Q8×C12, C12.26Q16
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), SD16, Q16, C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, Q8⋊C4, C4≀C2, C2×C3⋊C8, C4.Dic3, Q82S3, C3⋊Q16, C6.D4, Q8⋊C8, C12.55D4, Q82Dic3, Q83Dic3, C12.26Q16

Smallest permutation representation of C12.26Q16
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 15 30 58 172 71 88 106)(2 20 31 51 173 64 89 99)(3 13 32 56 174 69 90 104)(4 18 33 49 175 62 91 97)(5 23 34 54 176 67 92 102)(6 16 35 59 177 72 93 107)(7 21 36 52 178 65 94 100)(8 14 25 57 179 70 95 105)(9 19 26 50 180 63 96 98)(10 24 27 55 169 68 85 103)(11 17 28 60 170 61 86 108)(12 22 29 53 171 66 87 101)(37 128 151 118 166 138 183 73)(38 121 152 111 167 143 184 78)(39 126 153 116 168 136 185 83)(40 131 154 109 157 141 186 76)(41 124 155 114 158 134 187 81)(42 129 156 119 159 139 188 74)(43 122 145 112 160 144 189 79)(44 127 146 117 161 137 190 84)(45 132 147 110 162 142 191 77)(46 125 148 115 163 135 192 82)(47 130 149 120 164 140 181 75)(48 123 150 113 165 133 182 80)
(1 126 178 142)(2 127 179 143)(3 128 180 144)(4 129 169 133)(5 130 170 134)(6 131 171 135)(7 132 172 136)(8 121 173 137)(9 122 174 138)(10 123 175 139)(11 124 176 140)(12 125 177 141)(13 46 63 157)(14 47 64 158)(15 48 65 159)(16 37 66 160)(17 38 67 161)(18 39 68 162)(19 40 69 163)(20 41 70 164)(21 42 71 165)(22 43 72 166)(23 44 61 167)(24 45 62 168)(25 84 89 111)(26 73 90 112)(27 74 91 113)(28 75 92 114)(29 76 93 115)(30 77 94 116)(31 78 95 117)(32 79 96 118)(33 80 85 119)(34 81 86 120)(35 82 87 109)(36 83 88 110)(49 191 103 153)(50 192 104 154)(51 181 105 155)(52 182 106 156)(53 183 107 145)(54 184 108 146)(55 185 97 147)(56 186 98 148)(57 187 99 149)(58 188 100 150)(59 189 101 151)(60 190 102 152)```

`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), (1,15,30,58,172,71,88,106)(2,20,31,51,173,64,89,99)(3,13,32,56,174,69,90,104)(4,18,33,49,175,62,91,97)(5,23,34,54,176,67,92,102)(6,16,35,59,177,72,93,107)(7,21,36,52,178,65,94,100)(8,14,25,57,179,70,95,105)(9,19,26,50,180,63,96,98)(10,24,27,55,169,68,85,103)(11,17,28,60,170,61,86,108)(12,22,29,53,171,66,87,101)(37,128,151,118,166,138,183,73)(38,121,152,111,167,143,184,78)(39,126,153,116,168,136,185,83)(40,131,154,109,157,141,186,76)(41,124,155,114,158,134,187,81)(42,129,156,119,159,139,188,74)(43,122,145,112,160,144,189,79)(44,127,146,117,161,137,190,84)(45,132,147,110,162,142,191,77)(46,125,148,115,163,135,192,82)(47,130,149,120,164,140,181,75)(48,123,150,113,165,133,182,80), (1,126,178,142)(2,127,179,143)(3,128,180,144)(4,129,169,133)(5,130,170,134)(6,131,171,135)(7,132,172,136)(8,121,173,137)(9,122,174,138)(10,123,175,139)(11,124,176,140)(12,125,177,141)(13,46,63,157)(14,47,64,158)(15,48,65,159)(16,37,66,160)(17,38,67,161)(18,39,68,162)(19,40,69,163)(20,41,70,164)(21,42,71,165)(22,43,72,166)(23,44,61,167)(24,45,62,168)(25,84,89,111)(26,73,90,112)(27,74,91,113)(28,75,92,114)(29,76,93,115)(30,77,94,116)(31,78,95,117)(32,79,96,118)(33,80,85,119)(34,81,86,120)(35,82,87,109)(36,83,88,110)(49,191,103,153)(50,192,104,154)(51,181,105,155)(52,182,106,156)(53,183,107,145)(54,184,108,146)(55,185,97,147)(56,186,98,148)(57,187,99,149)(58,188,100,150)(59,189,101,151)(60,190,102,152)>;`

`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), (1,15,30,58,172,71,88,106)(2,20,31,51,173,64,89,99)(3,13,32,56,174,69,90,104)(4,18,33,49,175,62,91,97)(5,23,34,54,176,67,92,102)(6,16,35,59,177,72,93,107)(7,21,36,52,178,65,94,100)(8,14,25,57,179,70,95,105)(9,19,26,50,180,63,96,98)(10,24,27,55,169,68,85,103)(11,17,28,60,170,61,86,108)(12,22,29,53,171,66,87,101)(37,128,151,118,166,138,183,73)(38,121,152,111,167,143,184,78)(39,126,153,116,168,136,185,83)(40,131,154,109,157,141,186,76)(41,124,155,114,158,134,187,81)(42,129,156,119,159,139,188,74)(43,122,145,112,160,144,189,79)(44,127,146,117,161,137,190,84)(45,132,147,110,162,142,191,77)(46,125,148,115,163,135,192,82)(47,130,149,120,164,140,181,75)(48,123,150,113,165,133,182,80), (1,126,178,142)(2,127,179,143)(3,128,180,144)(4,129,169,133)(5,130,170,134)(6,131,171,135)(7,132,172,136)(8,121,173,137)(9,122,174,138)(10,123,175,139)(11,124,176,140)(12,125,177,141)(13,46,63,157)(14,47,64,158)(15,48,65,159)(16,37,66,160)(17,38,67,161)(18,39,68,162)(19,40,69,163)(20,41,70,164)(21,42,71,165)(22,43,72,166)(23,44,61,167)(24,45,62,168)(25,84,89,111)(26,73,90,112)(27,74,91,113)(28,75,92,114)(29,76,93,115)(30,77,94,116)(31,78,95,117)(32,79,96,118)(33,80,85,119)(34,81,86,120)(35,82,87,109)(36,83,88,110)(49,191,103,153)(50,192,104,154)(51,181,105,155)(52,182,106,156)(53,183,107,145)(54,184,108,146)(55,185,97,147)(56,186,98,148)(57,187,99,149)(58,188,100,150)(59,189,101,151)(60,190,102,152) );`

`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)], [(1,15,30,58,172,71,88,106),(2,20,31,51,173,64,89,99),(3,13,32,56,174,69,90,104),(4,18,33,49,175,62,91,97),(5,23,34,54,176,67,92,102),(6,16,35,59,177,72,93,107),(7,21,36,52,178,65,94,100),(8,14,25,57,179,70,95,105),(9,19,26,50,180,63,96,98),(10,24,27,55,169,68,85,103),(11,17,28,60,170,61,86,108),(12,22,29,53,171,66,87,101),(37,128,151,118,166,138,183,73),(38,121,152,111,167,143,184,78),(39,126,153,116,168,136,185,83),(40,131,154,109,157,141,186,76),(41,124,155,114,158,134,187,81),(42,129,156,119,159,139,188,74),(43,122,145,112,160,144,189,79),(44,127,146,117,161,137,190,84),(45,132,147,110,162,142,191,77),(46,125,148,115,163,135,192,82),(47,130,149,120,164,140,181,75),(48,123,150,113,165,133,182,80)], [(1,126,178,142),(2,127,179,143),(3,128,180,144),(4,129,169,133),(5,130,170,134),(6,131,171,135),(7,132,172,136),(8,121,173,137),(9,122,174,138),(10,123,175,139),(11,124,176,140),(12,125,177,141),(13,46,63,157),(14,47,64,158),(15,48,65,159),(16,37,66,160),(17,38,67,161),(18,39,68,162),(19,40,69,163),(20,41,70,164),(21,42,71,165),(22,43,72,166),(23,44,61,167),(24,45,62,168),(25,84,89,111),(26,73,90,112),(27,74,91,113),(28,75,92,114),(29,76,93,115),(30,77,94,116),(31,78,95,117),(32,79,96,118),(33,80,85,119),(34,81,86,120),(35,82,87,109),(36,83,88,110),(49,191,103,153),(50,192,104,154),(51,181,105,155),(52,182,106,156),(53,183,107,145),(54,184,108,146),(55,185,97,147),(56,186,98,148),(57,187,99,149),(58,188,100,150),(59,189,101,151),(60,190,102,152)]])`

48 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 8A ··· 8H 8I 8J 8K 8L 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 ··· 8 8 8 8 8 12 12 12 12 12 ··· 12 size 1 1 1 1 2 1 1 1 1 2 2 2 2 4 4 4 4 2 2 2 6 ··· 6 12 12 12 12 2 2 2 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - - - + - image C1 C2 C2 C2 C4 C4 C8 S3 D4 D6 Dic3 Dic3 M4(2) SD16 Q16 C3⋊D4 C3⋊C8 C4≀C2 C4.Dic3 Q8⋊2S3 C3⋊Q16 Q8⋊3Dic3 kernel C12.26Q16 C4×C3⋊C8 C12⋊C8 Q8×C12 C3×C4⋊C4 C6×Q8 C3×Q8 C4×Q8 C2×C12 C42 C4⋊C4 C2×Q8 C12 C12 C12 C2×C4 Q8 C6 C4 C4 C4 C2 # reps 1 1 1 1 2 2 8 1 2 1 1 1 2 2 2 4 4 4 4 1 1 2

Matrix representation of C12.26Q16 in GL4(𝔽73) generated by

 27 46 0 0 27 0 0 0 0 0 72 0 0 0 0 72
,
 11 39 0 0 50 62 0 0 0 0 67 6 0 0 67 67
,
 43 60 0 0 13 30 0 0 0 0 62 43 0 0 43 11
`G:=sub<GL(4,GF(73))| [27,27,0,0,46,0,0,0,0,0,72,0,0,0,0,72],[11,50,0,0,39,62,0,0,0,0,67,67,0,0,6,67],[43,13,0,0,60,30,0,0,0,0,62,43,0,0,43,11] >;`

C12.26Q16 in GAP, Magma, Sage, TeX

`C_{12}._{26}Q_{16}`
`% in TeX`

`G:=Group("C12.26Q16");`
`// GroupNames label`

`G:=SmallGroup(192,94);`
`// by ID`

`G=gap.SmallGroup(192,94);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,100,1123,570,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^8=1,c^2=a^6*b^4,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^9*b^-1>;`
`// generators/relations`

׿
×
𝔽