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## G = C6.5Q32order 192 = 26·3

### 3rd non-split extension by C6 of Q32 acting via Q32/Q16=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C6.5Q32
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C24⋊1C4 — C6.5Q32
 Lower central C3 — C6 — C12 — C24 — C6.5Q32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for C6.5Q32
G = < a,b,c | a6=b16=1, c2=b8, bab-1=a-1, ac=ca, cbc-1=a3b-1 >

Subgroups: 152 in 58 conjugacy classes, 31 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, Q16, C2×Q8, C24, C2×Dic3, C2×C12, C2×C12, C3×Q8, C2.D8, C2×C16, C2×Q16, C3⋊C16, C4⋊Dic3, C2×C24, C3×Q16, C3×Q16, C6×Q8, C2.Q32, C2×C3⋊C16, C241C4, C6×Q16, C6.5Q32
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C2×Dic3, C3⋊D4, D4⋊C4, SD32, Q32, D4⋊S3, D4.S3, C6.D4, C2.Q32, C8.6D6, C3⋊Q32, D4⋊Dic3, C6.5Q32

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

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

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

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

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 Dic3 SD16 D8 C3⋊D4 C3⋊D4 SD32 Q32 D4.S3 D4⋊S3 C8.6D6 C3⋊Q32 kernel C6.5Q32 C2×C3⋊C16 C24⋊1C4 C6×Q16 C3×Q16 C2×Q16 C24 C2×C12 C2×C8 Q16 C12 C2×C6 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 C6.5Q32 in GL4(𝔽97) generated by

 1 1 0 0 96 0 0 0 0 0 1 0 0 0 0 1
,
 89 36 0 0 44 8 0 0 0 0 26 95 0 0 2 26
,
 96 0 0 0 0 96 0 0 0 0 15 68 0 0 68 82
`G:=sub<GL(4,GF(97))| [1,96,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[89,44,0,0,36,8,0,0,0,0,26,2,0,0,95,26],[96,0,0,0,0,96,0,0,0,0,15,68,0,0,68,82] >;`

C6.5Q32 in GAP, Magma, Sage, TeX

`C_6._5Q_{32}`
`% in TeX`

`G:=Group("C6.5Q32");`
`// GroupNames label`

`G:=SmallGroup(192,123);`
`// by ID`

`G=gap.SmallGroup(192,123);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,675,346,192,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=b^16=1,c^2=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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