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## G = C6.Q32order 192 = 26·3

### 2nd non-split extension by C6 of Q32 acting via Q32/Q16=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C6.Q32
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C2×Dic12 — C6.Q32
 Lower central C3 — C6 — C12 — C24 — C6.Q32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2.D8

Generators and relations for C6.Q32
G = < a,b,c | a6=b16=1, c2=a3b8, bab-1=a-1, ac=ca, cbc-1=a3b-1 >

Subgroups: 176 in 58 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C16, C4⋊C4, C2×C8, Q16, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C2.D8, C2×C16, C2×Q16, C3⋊C16, Dic12, Dic12, C3×C4⋊C4, C2×C24, C2×Dic6, C2.Q32, C2×C3⋊C16, C3×C2.D8, C2×Dic12, C6.Q32
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, SD32, Q32, D6⋊C4, D4⋊S3, Q82S3, C2.Q32, C6.D8, D8.S3, C3⋊Q32, C6.Q32

Smallest permutation representation of C6.Q32
Regular action on 192 points
Generators in S192
```(1 190 143 153 82 100)(2 101 83 154 144 191)(3 192 129 155 84 102)(4 103 85 156 130 177)(5 178 131 157 86 104)(6 105 87 158 132 179)(7 180 133 159 88 106)(8 107 89 160 134 181)(9 182 135 145 90 108)(10 109 91 146 136 183)(11 184 137 147 92 110)(12 111 93 148 138 185)(13 186 139 149 94 112)(14 97 95 150 140 187)(15 188 141 151 96 98)(16 99 81 152 142 189)(17 42 122 64 73 169)(18 170 74 49 123 43)(19 44 124 50 75 171)(20 172 76 51 125 45)(21 46 126 52 77 173)(22 174 78 53 127 47)(23 48 128 54 79 175)(24 176 80 55 113 33)(25 34 114 56 65 161)(26 162 66 57 115 35)(27 36 116 58 67 163)(28 164 68 59 117 37)(29 38 118 60 69 165)(30 166 70 61 119 39)(31 40 120 62 71 167)(32 168 72 63 121 41)
(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 39 145 78)(2 69 146 46)(3 37 147 76)(4 67 148 44)(5 35 149 74)(6 65 150 42)(7 33 151 72)(8 79 152 40)(9 47 153 70)(10 77 154 38)(11 45 155 68)(12 75 156 36)(13 43 157 66)(14 73 158 34)(15 41 159 80)(16 71 160 48)(17 179 56 95)(18 86 57 186)(19 177 58 93)(20 84 59 184)(21 191 60 91)(22 82 61 182)(23 189 62 89)(24 96 63 180)(25 187 64 87)(26 94 49 178)(27 185 50 85)(28 92 51 192)(29 183 52 83)(30 90 53 190)(31 181 54 81)(32 88 55 188)(97 169 132 114)(98 121 133 176)(99 167 134 128)(100 119 135 174)(101 165 136 126)(102 117 137 172)(103 163 138 124)(104 115 139 170)(105 161 140 122)(106 113 141 168)(107 175 142 120)(108 127 143 166)(109 173 144 118)(110 125 129 164)(111 171 130 116)(112 123 131 162)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 16A ··· 16H 24A 24B 24C 24D order 1 2 2 2 3 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 16 ··· 16 24 24 24 24 size 1 1 1 1 2 2 2 8 8 24 24 2 2 2 2 2 2 2 4 4 8 8 8 8 6 ··· 6 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + + - - image C1 C2 C2 C2 C4 S3 D4 D4 D6 SD16 D8 C4×S3 D12 C3⋊D4 SD32 Q32 Q8⋊2S3 D4⋊S3 D8.S3 C3⋊Q32 kernel C6.Q32 C2×C3⋊C16 C3×C2.D8 C2×Dic12 Dic12 C2.D8 C24 C2×C12 C2×C8 C12 C2×C6 C8 C8 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 2 4 4 1 1 2 2

Matrix representation of C6.Q32 in GL4(𝔽97) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 96 1
,
 71 2 0 0 95 71 0 0 0 0 80 50 0 0 33 17
,
 60 52 0 0 52 37 0 0 0 0 75 0 0 0 0 75
`G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,0,96,0,0,1,1],[71,95,0,0,2,71,0,0,0,0,80,33,0,0,50,17],[60,52,0,0,52,37,0,0,0,0,75,0,0,0,0,75] >;`

C6.Q32 in GAP, Magma, Sage, TeX

`C_6.Q_{32}`
`% in TeX`

`G:=Group("C6.Q32");`
`// GroupNames label`

`G:=SmallGroup(192,51);`
`// by ID`

`G=gap.SmallGroup(192,51);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,675,346,192,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=b^16=1,c^2=a^3*b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;`
`// generators/relations`

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