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## G = C12.39SD16order 192 = 26·3

### 5th non-split extension by C12 of SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.39SD16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×C12 — C4×C3⋊C8 — C12.39SD16
 Lower central C3 — C6 — C12 — C12.39SD16
 Upper central C1 — C2×C4 — C42 — C4⋊C8

Generators and relations for C12.39SD16
G = < a,b,c | a12=b8=1, c2=a9, bab-1=a5, ac=ca, cbc-1=b3 >

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + - - + image C1 C2 C2 C2 C4 C8 S3 D4 Q8 D6 M4(2) SD16 Dic6 C4×S3 C3⋊D4 C8.C4 S3×C8 C8⋊S3 D4.S3 Q8⋊2S3 C12.53D4 kernel C12.39SD16 C4×C3⋊C8 C12⋊C8 C3×C4⋊C8 C2×C3⋊C8 C3⋊C8 C4⋊C8 C2×C12 C2×C12 C42 C12 C12 C2×C4 C2×C4 C2×C4 C6 C4 C4 C4 C4 C2 # reps 1 1 1 1 4 8 1 1 1 1 2 4 2 2 2 4 4 4 1 1 2

Matrix representation of C12.39SD16 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 27 46 0 0 27 0
,
 0 12 0 0 67 12 0 0 0 0 25 29 0 0 54 48
,
 50 58 0 0 6 23 0 0 0 0 51 0 0 0 0 51
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,27,27,0,0,46,0],[0,67,0,0,12,12,0,0,0,0,25,54,0,0,29,48],[50,6,0,0,58,23,0,0,0,0,51,0,0,0,0,51] >;

C12.39SD16 in GAP, Magma, Sage, TeX

C_{12}._{39}{\rm SD}_{16}
% in TeX

G:=Group("C12.39SD16");
// GroupNames label

G:=SmallGroup(192,39);
// by ID

G=gap.SmallGroup(192,39);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,589,36,100,570,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=1,c^2=a^9,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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