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G = C42.2D6order 192 = 26·3

2nd non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.2D6, C6.6C4≀C2, C8⋊C4.3S3, C4⋊Dic3.1C4, (C2×C4).106D12, (C2×C12).224D4, (C4×C12).11C22, C12.6Q8.4C2, C2.7(C424S3), C2.5(D12⋊C4), C22.57(D6⋊C4), C6.1(C4.10D4), C42.S3.1C2, C2.3(C12.47D4), C31(C42.2C22), (C2×C4).11(C4×S3), (C3×C8⋊C4).7C2, (C2×C12).23(C2×C4), (C2×C4).207(C3⋊D4), (C2×C6).38(C22⋊C4), SmallGroup(192,24)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.2D6
C1C3C6C2×C6C2×C12C4×C12C12.6Q8 — C42.2D6
C3C2×C6C2×C12 — C42.2D6
C1C22C42C8⋊C4

Generators and relations for C42.2D6
 G = < a,b,c,d | a4=b4=1, c6=a-1, d2=ba=ab, ac=ca, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=bc5 >

Subgroups: 152 in 60 conjugacy classes, 25 normal (all characteristic)
C1, C2 [×3], C3, C4 [×5], C22, C6 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×2], Dic3 [×2], C12 [×3], C2×C6, C42, C4⋊C4 [×4], C2×C8 [×2], C3⋊C8 [×2], C24 [×2], C2×Dic3 [×2], C2×C12 [×3], C8⋊C4, C8⋊C4, C42.C2, C2×C3⋊C8, Dic3⋊C4 [×2], C4⋊Dic3 [×2], C4×C12, C2×C24, C42.2C22, C42.S3, C3×C8⋊C4, C12.6Q8, C42.2D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, C4×S3, D12, C3⋊D4, C4.10D4, C4≀C2 [×2], D6⋊C4, C42.2C22, C424S3, C12.47D4, D12⋊C4, C42.2D6

Smallest permutation representation of C42.2D6
Regular action on 192 points
Generators in S192
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 84)(97 115 109 103)(98 116 110 104)(99 117 111 105)(100 118 112 106)(101 119 113 107)(102 120 114 108)(121 139 133 127)(122 140 134 128)(123 141 135 129)(124 142 136 130)(125 143 137 131)(126 144 138 132)(145 163 157 151)(146 164 158 152)(147 165 159 153)(148 166 160 154)(149 167 161 155)(150 168 162 156)(169 187 181 175)(170 188 182 176)(171 189 183 177)(172 190 184 178)(173 191 185 179)(174 192 186 180)
(1 89 39 72)(2 78 40 61)(3 91 41 50)(4 80 42 63)(5 93 43 52)(6 82 44 65)(7 95 45 54)(8 84 46 67)(9 73 47 56)(10 86 48 69)(11 75 25 58)(12 88 26 71)(13 77 27 60)(14 90 28 49)(15 79 29 62)(16 92 30 51)(17 81 31 64)(18 94 32 53)(19 83 33 66)(20 96 34 55)(21 85 35 68)(22 74 36 57)(23 87 37 70)(24 76 38 59)(97 167 131 183)(98 156 132 172)(99 145 133 185)(100 158 134 174)(101 147 135 187)(102 160 136 176)(103 149 137 189)(104 162 138 178)(105 151 139 191)(106 164 140 180)(107 153 141 169)(108 166 142 182)(109 155 143 171)(110 168 144 184)(111 157 121 173)(112 146 122 186)(113 159 123 175)(114 148 124 188)(115 161 125 177)(116 150 126 190)(117 163 127 179)(118 152 128 192)(119 165 129 181)(120 154 130 170)
(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)
(1 115 83 155 27 137 54 183)(2 154 96 124 28 182 67 102)(3 123 85 169 29 101 56 165)(4 192 74 112 30 164 69 134)(5 111 87 151 31 133 58 179)(6 150 76 144 32 178 71 98)(7 143 89 189 33 97 60 161)(8 188 78 108 34 160 49 130)(9 107 91 147 35 129 62 175)(10 146 80 140 36 174 51 118)(11 139 93 185 37 117 64 157)(12 184 82 104 38 156 53 126)(13 103 95 167 39 125 66 171)(14 166 84 136 40 170 55 114)(15 135 73 181 41 113 68 153)(16 180 86 100 42 152 57 122)(17 99 75 163 43 121 70 191)(18 162 88 132 44 190 59 110)(19 131 77 177 45 109 72 149)(20 176 90 120 46 148 61 142)(21 119 79 159 47 141 50 187)(22 158 92 128 48 186 63 106)(23 127 81 173 25 105 52 145)(24 172 94 116 26 168 65 138)

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

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

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

39 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D12E12F12G12H24A···24H
order12223444444466688888888121212121212121224···24
size11112222242424222444412121212222244444···4

39 irreducible representations

dim1111122222222444
type++++++++--
imageC1C2C2C2C4S3D4D6C4×S3D12C3⋊D4C4≀C2C424S3C4.10D4C12.47D4D12⋊C4
kernelC42.2D6C42.S3C3×C8⋊C4C12.6Q8C4⋊Dic3C8⋊C4C2×C12C42C2×C4C2×C4C2×C4C6C2C6C2C2
# reps1111412122288122

Matrix representation of C42.2D6 in GL4(𝔽73) generated by

306000
134300
00460
00046
,
71400
596600
00460
002127
,
396200
115000
00623
004811
,
12800
277200
00270
00711
G:=sub<GL(4,GF(73))| [30,13,0,0,60,43,0,0,0,0,46,0,0,0,0,46],[7,59,0,0,14,66,0,0,0,0,46,21,0,0,0,27],[39,11,0,0,62,50,0,0,0,0,62,48,0,0,3,11],[1,27,0,0,28,72,0,0,0,0,27,71,0,0,0,1] >;

C42.2D6 in GAP, Magma, Sage, TeX

C_4^2._2D_6
% in TeX

G:=Group("C4^2.2D6");
// GroupNames label

G:=SmallGroup(192,24);
// by ID

G=gap.SmallGroup(192,24);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,184,1571,570,192,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^-1,d^2=b*a=a*b,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=b*c^5>;
// generators/relations

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