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G = C42.8D6order 192 = 26·3

8th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.8D6, C6.19C4≀C2, C4⋊C4.1Dic3, (C2×C12).232D4, C42.C2.1S3, (C4×C12).236C22, C6.8(C4.10D4), C42.S3.9C2, C2.7(Q83Dic3), C2.3(C12.10D4), C32(C42.2C22), C22.40(C6.D4), (C3×C4⋊C4).1C4, (C2×C12).170(C2×C4), (C2×C4).10(C2×Dic3), (C3×C42.C2).7C2, (C2×C4).166(C3⋊D4), (C2×C6).101(C22⋊C4), SmallGroup(192,102)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.8D6
C1C3C6C2×C6C2×C12C4×C12C42.S3 — C42.8D6
C3C2×C6C2×C12 — C42.8D6
C1C22C42C42.C2

Generators and relations for C42.8D6
 G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a-1b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=a2bc5 >

Subgroups: 128 in 60 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2 [×2], C3, C4 [×5], C22, C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], C12 [×5], C2×C6, C42, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C3⋊C8 [×4], C2×C12, C2×C12 [×2], C2×C12 [×2], C8⋊C4 [×2], C42.C2, C2×C3⋊C8 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C42.2C22, C42.S3 [×2], C3×C42.C2, C42.8D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×Dic3, C3⋊D4 [×2], C4.10D4, C4≀C2 [×2], C6.D4, C42.2C22, C12.10D4, Q83Dic3 [×2], C42.8D6

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

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

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

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

33 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G6A6B6C8A···8H12A···12F12G12H12I12J
order1222344444446668···812···1212121212
size11112222248822212···124···48888

33 irreducible representations

dim1111222222444
type++++++--
imageC1C2C2C4S3D4D6Dic3C3⋊D4C4≀C2C4.10D4C12.10D4Q83Dic3
kernelC42.8D6C42.S3C3×C42.C2C3×C4⋊C4C42.C2C2×C12C42C4⋊C4C2×C4C6C6C2C2
# reps1214121248124

Matrix representation of C42.8D6 in GL6(𝔽73)

0270000
4600000
001000
000100
0000460
0000046
,
010000
7200000
001000
000100
000001
0000720
,
11430000
43620000
0007200
0017200
00004467
00006729
,
10470000
47630000
00513200
00102200
00001360
00001313

G:=sub<GL(6,GF(73))| [0,46,0,0,0,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[11,43,0,0,0,0,43,62,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,44,67,0,0,0,0,67,29],[10,47,0,0,0,0,47,63,0,0,0,0,0,0,51,10,0,0,0,0,32,22,0,0,0,0,0,0,13,13,0,0,0,0,60,13] >;

C42.8D6 in GAP, Magma, Sage, TeX

C_4^2._8D_6
% in TeX

G:=Group("C4^2.8D6");
// GroupNames label

G:=SmallGroup(192,102);
// by ID

G=gap.SmallGroup(192,102);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,120,219,268,1571,570,136,6278]);
// Polycyclic

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

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