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G = (C2×C12).54D4order 192 = 26·3

28th non-split extension by C2×C12 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C12).54D4, C6.22(C4⋊Q8), (C2×C12).39Q8, C2.15(C12⋊Q8), (C2×Dic3).6Q8, (C2×C4).12Dic6, C22.47(S3×Q8), C6.89(C4⋊D4), C2.6(D63Q8), (C2×Dic3).59D4, C22.245(S3×D4), (C22×C4).116D6, C6.59(C22⋊Q8), C2.8(C4.Dic6), C6.21(C42.C2), C22.48(C2×Dic6), C2.19(D6.D4), C2.12(Dic3.Q8), C6.C42.18C2, C23.386(C22×S3), (C22×C12).65C22, (C22×C6).346C23, C2.12(C23.14D6), C2.10(C12.48D4), C22.104(C4○D12), C34(C23.81C23), C22.48(Q83S3), C6.50(C22.D4), C22.100(D42S3), (C22×Dic3).54C22, (C6×C4⋊C4).23C2, (C2×C4⋊C4).19S3, (C2×C6).37(C2×Q8), (C2×C6).332(C2×D4), (C2×C6).85(C4○D4), (C2×C4).37(C3⋊D4), (C2×C4⋊Dic3).19C2, (C2×Dic3⋊C4).14C2, C22.136(C2×C3⋊D4), SmallGroup(192,541)

Series: Derived Chief Lower central Upper central

C1C22×C6 — (C2×C12).54D4
C1C3C6C2×C6C22×C6C22×Dic3C2×Dic3⋊C4 — (C2×C12).54D4
C3C22×C6 — (C2×C12).54D4
C1C23C2×C4⋊C4

Generators and relations for (C2×C12).54D4
 G = < a,b,c,d | a2=b12=c4=1, d2=ab6, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=b6c-1 >

Subgroups: 360 in 150 conjugacy classes, 63 normal (51 characteristic)
C1, C2 [×7], C3, C4 [×11], C22 [×7], C6 [×7], C2×C4 [×4], C2×C4 [×21], C23, Dic3 [×6], C12 [×5], C2×C6 [×7], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×4], C2×C12 [×7], C22×C6, C2.C42 [×3], C2×C4⋊C4, C2×C4⋊C4 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×3], C23.81C23, C6.C42 [×3], C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C6×C4⋊C4, (C2×C12).54D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], Q8 [×4], C23, D6 [×3], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], Dic6 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×Dic6, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C23.81C23, C12⋊Q8, Dic3.Q8, C4.Dic6, D6.D4, C12.48D4, C23.14D6, D63Q8, (C2×C12).54D4

Smallest permutation representation of (C2×C12).54D4
Regular action on 192 points
Generators in S192
(1 69)(2 70)(3 71)(4 72)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 97)(21 98)(22 99)(23 100)(24 101)(25 162)(26 163)(27 164)(28 165)(29 166)(30 167)(31 168)(32 157)(33 158)(34 159)(35 160)(36 161)(37 94)(38 95)(39 96)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 187)(50 188)(51 189)(52 190)(53 191)(54 192)(55 181)(56 182)(57 183)(58 184)(59 185)(60 186)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(81 155)(82 156)(83 145)(84 146)(109 131)(110 132)(111 121)(112 122)(113 123)(114 124)(115 125)(116 126)(117 127)(118 128)(119 129)(120 130)(133 171)(134 172)(135 173)(136 174)(137 175)(138 176)(139 177)(140 178)(141 179)(142 180)(143 169)(144 170)
(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)
(1 146 20 181)(2 145 21 192)(3 156 22 191)(4 155 23 190)(5 154 24 189)(6 153 13 188)(7 152 14 187)(8 151 15 186)(9 150 16 185)(10 149 17 184)(11 148 18 183)(12 147 19 182)(25 47 124 138)(26 46 125 137)(27 45 126 136)(28 44 127 135)(29 43 128 134)(30 42 129 133)(31 41 130 144)(32 40 131 143)(33 39 132 142)(34 38 121 141)(35 37 122 140)(36 48 123 139)(49 63 78 103)(50 62 79 102)(51 61 80 101)(52 72 81 100)(53 71 82 99)(54 70 83 98)(55 69 84 97)(56 68 73 108)(57 67 74 107)(58 66 75 106)(59 65 76 105)(60 64 77 104)(85 109 169 157)(86 120 170 168)(87 119 171 167)(88 118 172 166)(89 117 173 165)(90 116 174 164)(91 115 175 163)(92 114 176 162)(93 113 177 161)(94 112 178 160)(95 111 179 159)(96 110 180 158)
(1 29 63 160)(2 165 64 34)(3 27 65 158)(4 163 66 32)(5 25 67 168)(6 161 68 30)(7 35 69 166)(8 159 70 28)(9 33 71 164)(10 157 72 26)(11 31 61 162)(12 167 62 36)(13 113 108 129)(14 122 97 118)(15 111 98 127)(16 132 99 116)(17 109 100 125)(18 130 101 114)(19 119 102 123)(20 128 103 112)(21 117 104 121)(22 126 105 110)(23 115 106 131)(24 124 107 120)(37 49 88 181)(38 186 89 54)(39 59 90 191)(40 184 91 52)(41 57 92 189)(42 182 93 50)(43 55 94 187)(44 192 95 60)(45 53 96 185)(46 190 85 58)(47 51 86 183)(48 188 87 56)(73 139 153 171)(74 176 154 144)(75 137 155 169)(76 174 156 142)(77 135 145 179)(78 172 146 140)(79 133 147 177)(80 170 148 138)(81 143 149 175)(82 180 150 136)(83 141 151 173)(84 178 152 134)

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

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

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

42 conjugacy classes

class 1 2A···2G 3 4A···4F4G···4N6A···6G12A···12L
order12···234···44···46···612···12
size11···124···412···122···24···4

42 irreducible representations

dim1111122222222224444
type+++++++-+-+-+--+
imageC1C2C2C2C2S3D4Q8D4Q8D6C4○D4Dic6C3⋊D4C4○D12S3×D4D42S3S3×Q8Q83S3
kernel(C2×C12).54D4C6.C42C2×Dic3⋊C4C2×C4⋊Dic3C6×C4⋊C4C2×C4⋊C4C2×Dic3C2×Dic3C2×C12C2×C12C22×C4C2×C6C2×C4C2×C4C22C22C22C22C22
# reps1321112222364441111

Matrix representation of (C2×C12).54D4 in GL6(𝔽13)

100000
010000
0012000
0001200
000010
000001
,
720000
160000
0012000
000100
0000112
000010
,
150000
10120000
0012000
0001200
000058
000008
,
150000
10120000
000100
0012000
000033
0000610

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,1,0,0,0,0,2,6,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,10,0,0,0,0,5,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,8,8],[1,10,0,0,0,0,5,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,3,6,0,0,0,0,3,10] >;

(C2×C12).54D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})._{54}D_4
% in TeX

G:=Group("(C2xC12).54D4");
// GroupNames label

G:=SmallGroup(192,541);
// by ID

G=gap.SmallGroup(192,541);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,387,184,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=a*b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations

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