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G = C12.53D8order 192 = 26·3

7th non-split extension by C12 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.53D8, C12.24Q16, C42.190D6, C12.1M4(2), C3⋊C81C8, C4⋊C8.1S3, C32(C81C8), C6.6(C4⋊C8), C4.11(S3×C8), C12.1(C2×C8), (C2×C12).31Q8, C12⋊C8.6C2, C6.1(C2.D8), C4.7(C8⋊S3), C4.26(D4⋊S3), (C2×C12).485D4, (C2×C4).18Dic6, C6.4(C8.C4), C2.3(Dic3⋊C8), (C4×C12).37C22, C4.12(C3⋊Q16), C2.1(C6.Q16), C2.1(C12.53D4), C22.18(Dic3⋊C4), (C4×C3⋊C8).1C2, (C2×C3⋊C8).4C4, (C3×C4⋊C8).1C2, (C2×C6).31(C4⋊C4), (C2×C12).46(C2×C4), (C2×C4).133(C4×S3), (C2×C4).263(C3⋊D4), SmallGroup(192,38)

Series: Derived Chief Lower central Upper central

C1C12 — C12.53D8
C1C3C6C2×C6C2×C12C4×C12C4×C3⋊C8 — C12.53D8
C3C6C12 — C12.53D8
C1C2×C4C42C4⋊C8

Generators and relations for C12.53D8
 G = < a,b,c | a12=b8=1, c2=a9, bab-1=cac-1=a5, cbc-1=b-1 >

2C4
3C8
3C8
4C8
6C8
12C8
2C12
2C2×C8
3C2×C8
3C2×C8
6C2×C8
2C3⋊C8
4C24
4C3⋊C8
3C4⋊C8
3C4×C8
2C2×C24
2C2×C3⋊C8
3C81C8

Smallest permutation representation of C12.53D8
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)
(1 175 101 185 85 19 38 77)(2 180 102 190 86 24 39 82)(3 173 103 183 87 17 40 75)(4 178 104 188 88 22 41 80)(5 171 105 181 89 15 42 73)(6 176 106 186 90 20 43 78)(7 169 107 191 91 13 44 83)(8 174 108 184 92 18 45 76)(9 179 97 189 93 23 46 81)(10 172 98 182 94 16 47 74)(11 177 99 187 95 21 48 79)(12 170 100 192 96 14 37 84)(25 110 126 161 138 50 66 152)(26 115 127 166 139 55 67 145)(27 120 128 159 140 60 68 150)(28 113 129 164 141 53 69 155)(29 118 130 157 142 58 70 148)(30 111 131 162 143 51 71 153)(31 116 132 167 144 56 72 146)(32 109 121 160 133 49 61 151)(33 114 122 165 134 54 62 156)(34 119 123 158 135 59 63 149)(35 112 124 163 136 52 64 154)(36 117 125 168 137 57 65 147)
(1 159 10 168 7 165 4 162)(2 164 11 161 8 158 5 167)(3 157 12 166 9 163 6 160)(13 62 22 71 19 68 16 65)(14 67 23 64 20 61 17 70)(15 72 24 69 21 66 18 63)(25 184 34 181 31 190 28 187)(26 189 35 186 32 183 29 192)(27 182 36 191 33 188 30 185)(37 55 46 52 43 49 40 58)(38 60 47 57 44 54 41 51)(39 53 48 50 45 59 42 56)(73 144 82 141 79 138 76 135)(74 137 83 134 80 143 77 140)(75 142 84 139 81 136 78 133)(85 150 94 147 91 156 88 153)(86 155 95 152 92 149 89 146)(87 148 96 145 93 154 90 151)(97 112 106 109 103 118 100 115)(98 117 107 114 104 111 101 120)(99 110 108 119 105 116 102 113)(121 173 130 170 127 179 124 176)(122 178 131 175 128 172 125 169)(123 171 132 180 129 177 126 174)

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

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

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

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H6A6B6C8A8B8C8D8E···8L8M8N8O8P12A12B12C12D12E12F12G12H24A···24H
order122234444444466688888···88888121212121212121224···24
size111121111222222244446···612121212222244444···4

48 irreducible representations

dim1111112222222222222444
type++++++-++--+-
imageC1C2C2C2C4C8S3D4Q8D6M4(2)D8Q16Dic6C4×S3C3⋊D4C8.C4S3×C8C8⋊S3D4⋊S3C3⋊Q16C12.53D4
kernelC12.53D8C4×C3⋊C8C12⋊C8C3×C4⋊C8C2×C3⋊C8C3⋊C8C4⋊C8C2×C12C2×C12C42C12C12C12C2×C4C2×C4C2×C4C6C4C4C4C4C2
# reps1111481111222222444112

Matrix representation of C12.53D8 in GL4(𝔽73) generated by

274600
27000
0010
0001
,
513200
102200
004132
00570
,
462600
722700
001454
004159
G:=sub<GL(4,GF(73))| [27,27,0,0,46,0,0,0,0,0,1,0,0,0,0,1],[51,10,0,0,32,22,0,0,0,0,41,57,0,0,32,0],[46,72,0,0,26,27,0,0,0,0,14,41,0,0,54,59] >;

C12.53D8 in GAP, Magma, Sage, TeX

C_{12}._{53}D_8
% in TeX

G:=Group("C12.53D8");
// GroupNames label

G:=SmallGroup(192,38);
// by ID

G=gap.SmallGroup(192,38);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,141,36,100,570,136,6278]);
// Polycyclic

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

Export

Subgroup lattice of C12.53D8 in TeX

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