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## G = C6.SD32order 192 = 26·3

### 1st non-split extension by C6 of SD32 acting via SD32/Q16=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C6.SD32
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C2×C3⋊C16 — C6.SD32
 Lower central C3 — C6 — C12 — C24 — C6.SD32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2.D8

Generators and relations for C6.SD32
G = < a,b,c | a6=b16=1, c2=a3, bab-1=cac-1=a-1, cbc-1=b7 >

Smallest permutation representation of C6.SD32
Regular action on 192 points
Generators in S192
(1 73 126 44 23 191)(2 192 24 45 127 74)(3 75 128 46 25 177)(4 178 26 47 113 76)(5 77 114 48 27 179)(6 180 28 33 115 78)(7 79 116 34 29 181)(8 182 30 35 117 80)(9 65 118 36 31 183)(10 184 32 37 119 66)(11 67 120 38 17 185)(12 186 18 39 121 68)(13 69 122 40 19 187)(14 188 20 41 123 70)(15 71 124 42 21 189)(16 190 22 43 125 72)(49 174 149 139 103 87)(50 88 104 140 150 175)(51 176 151 141 105 89)(52 90 106 142 152 161)(53 162 153 143 107 91)(54 92 108 144 154 163)(55 164 155 129 109 93)(56 94 110 130 156 165)(57 166 157 131 111 95)(58 96 112 132 158 167)(59 168 159 133 97 81)(60 82 98 134 160 169)(61 170 145 135 99 83)(62 84 100 136 146 171)(63 172 147 137 101 85)(64 86 102 138 148 173)
(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 58 44 132)(2 49 45 139)(3 56 46 130)(4 63 47 137)(5 54 48 144)(6 61 33 135)(7 52 34 142)(8 59 35 133)(9 50 36 140)(10 57 37 131)(11 64 38 138)(12 55 39 129)(13 62 40 136)(14 53 41 143)(15 60 42 134)(16 51 43 141)(17 102 67 173)(18 109 68 164)(19 100 69 171)(20 107 70 162)(21 98 71 169)(22 105 72 176)(23 112 73 167)(24 103 74 174)(25 110 75 165)(26 101 76 172)(27 108 77 163)(28 99 78 170)(29 106 79 161)(30 97 80 168)(31 104 65 175)(32 111 66 166)(81 117 159 182)(82 124 160 189)(83 115 145 180)(84 122 146 187)(85 113 147 178)(86 120 148 185)(87 127 149 192)(88 118 150 183)(89 125 151 190)(90 116 152 181)(91 123 153 188)(92 114 154 179)(93 121 155 186)(94 128 156 177)(95 119 157 184)(96 126 158 191)

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

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

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

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 4 4 4 4 type + + + + + - + + - + - - + - + image C1 C2 C2 C2 C4 S3 Q8 D4 D6 Q16 D8 Dic6 C4×S3 C3⋊D4 SD32 C3⋊Q16 D4⋊S3 D8.S3 C8.6D6 kernel C6.SD32 C2×C3⋊C16 C24⋊1C4 C3×C2.D8 C3⋊C16 C2.D8 C24 C2×C12 C2×C8 C12 C2×C6 C8 C8 C2×C4 C6 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 2 8 1 1 2 2

Matrix representation of C6.SD32 in GL4(𝔽97) generated by

 0 1 0 0 96 1 0 0 0 0 1 0 0 0 0 1
,
 79 21 0 0 3 18 0 0 0 0 87 53 0 0 44 87
,
 80 50 0 0 33 17 0 0 0 0 63 94 0 0 94 34
G:=sub<GL(4,GF(97))| [0,96,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[79,3,0,0,21,18,0,0,0,0,87,44,0,0,53,87],[80,33,0,0,50,17,0,0,0,0,63,94,0,0,94,34] >;

C6.SD32 in GAP, Magma, Sage, TeX

C_6.{\rm SD}_{32}
% in TeX

G:=Group("C6.SD32");
// GroupNames label

G:=SmallGroup(192,49);
// by ID

G=gap.SmallGroup(192,49);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,589,36,346,192,851,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^6=b^16=1,c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^7>;
// generators/relations

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