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## G = C12.9C42order 192 = 26·3

### 2nd non-split extension by C12 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.9C42
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C2×C4⋊Dic3 — C12.9C42
 Lower central C3 — C6 — C12 — C12.9C42
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C12.9C42
G = < a,b,c | a12=b4=1, c4=a6, bab-1=a-1, ac=ca, cbc-1=a3b >

Subgroups: 296 in 114 conjugacy classes, 67 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C22×C8, C4⋊Dic3, C4⋊Dic3, C2×C24, C2×C24, C22×Dic3, C22×C12, C22.4Q16, C2×C4⋊Dic3, C22×C24, C12.9C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C24⋊C2, D24, Dic12, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C22.4Q16, C2.Dic12, C8⋊Dic3, C241C4, C2.D24, C6.C42, C12.9C42

Smallest permutation representation of C12.9C42
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)
(1 153 81 128)(2 152 82 127)(3 151 83 126)(4 150 84 125)(5 149 73 124)(6 148 74 123)(7 147 75 122)(8 146 76 121)(9 145 77 132)(10 156 78 131)(11 155 79 130)(12 154 80 129)(13 111 66 57)(14 110 67 56)(15 109 68 55)(16 120 69 54)(17 119 70 53)(18 118 71 52)(19 117 72 51)(20 116 61 50)(21 115 62 49)(22 114 63 60)(23 113 64 59)(24 112 65 58)(25 166 107 172)(26 165 108 171)(27 164 97 170)(28 163 98 169)(29 162 99 180)(30 161 100 179)(31 160 101 178)(32 159 102 177)(33 158 103 176)(34 157 104 175)(35 168 105 174)(36 167 106 173)(37 139 91 189)(38 138 92 188)(39 137 93 187)(40 136 94 186)(41 135 95 185)(42 134 96 184)(43 133 85 183)(44 144 86 182)(45 143 87 181)(46 142 88 192)(47 141 89 191)(48 140 90 190)
(1 14 173 183 7 20 179 189)(2 15 174 184 8 21 180 190)(3 16 175 185 9 22 169 191)(4 17 176 186 10 23 170 192)(5 18 177 187 11 24 171 181)(6 19 178 188 12 13 172 182)(25 47 154 114 31 41 148 120)(26 48 155 115 32 42 149 109)(27 37 156 116 33 43 150 110)(28 38 145 117 34 44 151 111)(29 39 146 118 35 45 152 112)(30 40 147 119 36 46 153 113)(49 102 96 124 55 108 90 130)(50 103 85 125 56 97 91 131)(51 104 86 126 57 98 92 132)(52 105 87 127 58 99 93 121)(53 106 88 128 59 100 94 122)(54 107 89 129 60 101 95 123)(61 161 139 81 67 167 133 75)(62 162 140 82 68 168 134 76)(63 163 141 83 69 157 135 77)(64 164 142 84 70 158 136 78)(65 165 143 73 71 159 137 79)(66 166 144 74 72 160 138 80)```

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E ··· 4L 6A ··· 6G 8A ··· 8H 12A ··· 12H 24A ··· 24P order 1 2 ··· 2 3 4 4 4 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 ··· 1 2 2 2 2 2 12 ··· 12 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + + - - + + - image C1 C2 C2 C4 C4 S3 D4 Q8 D4 Dic3 D6 D8 SD16 Q16 Dic6 C4×S3 C3⋊D4 D12 C24⋊C2 D24 Dic12 kernel C12.9C42 C2×C4⋊Dic3 C22×C24 C4⋊Dic3 C2×C24 C22×C8 C2×C12 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C6 C2×C6 C2×C4 C2×C4 C2×C4 C23 C22 C22 C22 # reps 1 2 1 8 4 1 2 1 1 2 1 2 4 2 2 4 4 2 8 4 4

Matrix representation of C12.9C42 in GL5(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 72 72 0 0 0 0 0 59 66 0 0 0 7 66
,
 1 0 0 0 0 0 11 48 0 0 0 37 62 0 0 0 0 0 53 55 0 0 0 2 20
,
 46 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 36 11 0 0 0 62 25

`G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,59,7,0,0,0,66,66],[1,0,0,0,0,0,11,37,0,0,0,48,62,0,0,0,0,0,53,2,0,0,0,55,20],[46,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,36,62,0,0,0,11,25] >;`

C12.9C42 in GAP, Magma, Sage, TeX

`C_{12}._9C_4^2`
`% in TeX`

`G:=Group("C12.9C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,110);`
`// by ID`

`G=gap.SmallGroup(192,110);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,176,1123,136,6278]);`
`// Polycyclic`

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

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