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G = C12.5Q16order 192 = 26·3

5th non-split extension by C12 of Q16 acting via Q16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12.5Q16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×C12 — C12⋊C8 — C12.5Q16
 Lower central C3 — C2×C6 — C2×C12 — C12.5Q16
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C12.5Q16
G = < a,b,c | a12=b8=1, c2=a6b4, bab-1=a-1, cac-1=a7, cbc-1=a3b-1 >

Subgroups: 144 in 64 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2 [×2], C3, C4 [×4], C4 [×3], C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], Q8 [×2], C12 [×4], C12 [×3], C2×C6, C42, C4⋊C4 [×2], C2×C8 [×2], C2×Q8 [×2], C3⋊C8 [×2], C2×C12, C2×C12 [×2], C2×C12 [×2], C3×Q8 [×2], C4⋊C8 [×2], C4⋊Q8, C2×C3⋊C8 [×2], C4×C12, C3×C4⋊C4 [×2], C6×Q8 [×2], C4.6Q16, C12⋊C8 [×2], C3×C4⋊Q8, C12.5Q16
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, SD16 [×2], Q16 [×2], C2×Dic3, C3⋊D4 [×2], C4.D4, Q8⋊C4 [×2], Q82S3 [×2], C3⋊Q16 [×2], C6.D4, C4.6Q16, C12.D4, Q82Dic3 [×2], C12.5Q16

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - - + + - image C1 C2 C2 C4 S3 D4 D6 Dic3 SD16 Q16 C3⋊D4 C4.D4 Q8⋊2S3 C3⋊Q16 C12.D4 kernel C12.5Q16 C12⋊C8 C3×C4⋊Q8 C6×Q8 C4⋊Q8 C2×C12 C42 C2×Q8 C12 C12 C2×C4 C6 C4 C4 C2 # reps 1 2 1 4 1 2 1 2 4 4 4 1 2 2 2

Matrix representation of C12.5Q16 in GL6(𝔽73)

 0 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 49 70 0 0 0 0 70 24 0 0 0 0 0 0 6 6 0 0 0 0 67 6 0 0 0 0 0 0 26 39 0 0 0 0 65 47
,
 53 16 0 0 0 0 16 20 0 0 0 0 0 0 54 21 0 0 0 0 21 19 0 0 0 0 0 0 30 13 0 0 0 0 60 43

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[49,70,0,0,0,0,70,24,0,0,0,0,0,0,6,67,0,0,0,0,6,6,0,0,0,0,0,0,26,65,0,0,0,0,39,47],[53,16,0,0,0,0,16,20,0,0,0,0,0,0,54,21,0,0,0,0,21,19,0,0,0,0,0,0,30,60,0,0,0,0,13,43] >;`

C12.5Q16 in GAP, Magma, Sage, TeX

`C_{12}._5Q_{16}`
`% in TeX`

`G:=Group("C12.5Q16");`
`// GroupNames label`

`G:=SmallGroup(192,105);`
`// by ID`

`G=gap.SmallGroup(192,105);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,219,100,1571,570,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^8=1,c^2=a^6*b^4,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b^-1>;`
`// generators/relations`

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