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## G = C12.SD16order 192 = 26·3

### 17th non-split extension by C12 of SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12.SD16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C4×C3⋊C8 — C12.SD16
 Lower central C3 — C6 — C2×C12 — C12.SD16
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C12.SD16
G = < a,b,c | a12=b8=1, c2=a6, bab-1=a5, cac-1=a7, cbc-1=b3 >

Subgroups: 240 in 98 conjugacy classes, 51 normal (23 characteristic)
C1, C2 [×3], C3, C4 [×6], C4 [×4], C22, C6 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×4], Dic3 [×2], C12 [×6], C12 [×2], C2×C6, C42, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×Q8 [×2], C3⋊C8 [×4], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C4×C8, C4.Q8 [×4], C4⋊Q8, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C6×Q8, C83Q8, C4×C3⋊C8, C12.Q8 [×4], C122Q8, C3×C4⋊Q8, C12.SD16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], SD16 [×4], C2×D4, C2×Q8 [×2], C3⋊D4 [×2], C22×S3, C4⋊Q8, C2×SD16 [×2], D4.S3 [×2], Q82S3 [×2], S3×Q8 [×2], C2×C3⋊D4, C83Q8, C2×D4.S3, C2×Q82S3, Dic3⋊Q8, C12.SD16

Smallest permutation representation of C12.SD16
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 48 177 120 108 184 139 129)(2 41 178 113 97 189 140 122)(3 46 179 118 98 182 141 127)(4 39 180 111 99 187 142 132)(5 44 169 116 100 192 143 125)(6 37 170 109 101 185 144 130)(7 42 171 114 102 190 133 123)(8 47 172 119 103 183 134 128)(9 40 173 112 104 188 135 121)(10 45 174 117 105 181 136 126)(11 38 175 110 106 186 137 131)(12 43 176 115 107 191 138 124)(13 64 33 87 77 57 161 145)(14 69 34 92 78 50 162 150)(15 62 35 85 79 55 163 155)(16 67 36 90 80 60 164 148)(17 72 25 95 81 53 165 153)(18 65 26 88 82 58 166 146)(19 70 27 93 83 51 167 151)(20 63 28 86 84 56 168 156)(21 68 29 91 73 49 157 149)(22 61 30 96 74 54 158 154)(23 66 31 89 75 59 159 147)(24 71 32 94 76 52 160 152)
(1 54 7 60)(2 49 8 55)(3 56 9 50)(4 51 10 57)(5 58 11 52)(6 53 12 59)(13 39 19 45)(14 46 20 40)(15 41 21 47)(16 48 22 42)(17 43 23 37)(18 38 24 44)(25 124 31 130)(26 131 32 125)(27 126 33 132)(28 121 34 127)(29 128 35 122)(30 123 36 129)(61 102 67 108)(62 97 68 103)(63 104 69 98)(64 99 70 105)(65 106 71 100)(66 101 72 107)(73 183 79 189)(74 190 80 184)(75 185 81 191)(76 192 82 186)(77 187 83 181)(78 182 84 188)(85 178 91 172)(86 173 92 179)(87 180 93 174)(88 175 94 169)(89 170 95 176)(90 177 96 171)(109 165 115 159)(110 160 116 166)(111 167 117 161)(112 162 118 168)(113 157 119 163)(114 164 120 158)(133 148 139 154)(134 155 140 149)(135 150 141 156)(136 145 142 151)(137 152 143 146)(138 147 144 153)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4F 4G 4H 4I 4J 6A 6B 6C 8A ··· 8H 12A ··· 12F 12G 12H 12I 12J order 1 2 2 2 3 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 8 8 24 24 2 2 2 6 ··· 6 4 ··· 4 8 8 8 8

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + - + - image C1 C2 C2 C2 C2 S3 Q8 D4 D6 D6 SD16 C3⋊D4 D4.S3 Q8⋊2S3 S3×Q8 kernel C12.SD16 C4×C3⋊C8 C12.Q8 C12⋊2Q8 C3×C4⋊Q8 C4⋊Q8 C3⋊C8 C2×C12 C42 C4⋊C4 C12 C2×C4 C4 C4 C4 # reps 1 1 4 1 1 1 4 2 1 2 8 4 2 2 2

Matrix representation of C12.SD16 in GL6(𝔽73)

 0 72 0 0 0 0 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 55 50 0 0 0 0 68 18 0 0 0 0 0 0 0 18 0 0 0 0 69 61 0 0 0 0 0 0 67 6 0 0 0 0 67 67
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 21 64 0 0 0 0 57 52 0 0 0 0 0 0 43 62 0 0 0 0 62 30

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[55,68,0,0,0,0,50,18,0,0,0,0,0,0,0,69,0,0,0,0,18,61,0,0,0,0,0,0,67,67,0,0,0,0,6,67],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,21,57,0,0,0,0,64,52,0,0,0,0,0,0,43,62,0,0,0,0,62,30] >;

C12.SD16 in GAP, Magma, Sage, TeX

C_{12}.{\rm SD}_{16}
% in TeX

G:=Group("C12.SD16");
// GroupNames label

G:=SmallGroup(192,639);
// by ID

G=gap.SmallGroup(192,639);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,1094,135,58,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=b^3>;
// generators/relations

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