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G = C6×C2.C42order 192 = 26·3

Direct product of C6 and C2.C42

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C6×C2.C42
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C3×C2.C42 — C6×C2.C42
 Lower central C1 — C2 — C6×C2.C42
 Upper central C1 — C23×C6 — C6×C2.C42

Generators and relations for C6×C2.C42
G = < a,b,c,d | a6=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 450 in 330 conjugacy classes, 210 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C23, C12, C2×C6, C2×C6, C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C2.C42, C23×C4, C22×C12, C22×C12, C23×C6, C2×C2.C42, C3×C2.C42, C23×C12, C6×C2.C42
Quotients:

Smallest permutation representation of C6×C2.C42
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 56)(2 57)(3 58)(4 59)(5 60)(6 55)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 65)(20 66)(21 61)(22 62)(23 63)(24 64)(25 74)(26 75)(27 76)(28 77)(29 78)(30 73)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)(43 150)(44 145)(45 146)(46 147)(47 148)(48 149)(49 124)(50 125)(51 126)(52 121)(53 122)(54 123)(67 80)(68 81)(69 82)(70 83)(71 84)(72 79)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 172)(98 173)(99 174)(100 169)(101 170)(102 171)(103 113)(104 114)(105 109)(106 110)(107 111)(108 112)(115 128)(116 129)(117 130)(118 131)(119 132)(120 127)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)(151 161)(152 162)(153 157)(154 158)(155 159)(156 160)(163 176)(164 177)(165 178)(166 179)(167 180)(168 175)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)
(1 133 68 171)(2 134 69 172)(3 135 70 173)(4 136 71 174)(5 137 72 169)(6 138 67 170)(7 86 42 124)(8 87 37 125)(9 88 38 126)(10 89 39 121)(11 90 40 122)(12 85 41 123)(13 92 32 49)(14 93 33 50)(15 94 34 51)(16 95 35 52)(17 96 36 53)(18 91 31 54)(19 154 28 179)(20 155 29 180)(21 156 30 175)(22 151 25 176)(23 152 26 177)(24 153 27 178)(43 114 181 116)(44 109 182 117)(45 110 183 118)(46 111 184 119)(47 112 185 120)(48 113 186 115)(55 144 80 101)(56 139 81 102)(57 140 82 97)(58 141 83 98)(59 142 84 99)(60 143 79 100)(61 160 73 168)(62 161 74 163)(63 162 75 164)(64 157 76 165)(65 158 77 166)(66 159 78 167)(103 192 128 149)(104 187 129 150)(105 188 130 145)(106 189 131 146)(107 190 132 147)(108 191 127 148)
(1 132 20 123)(2 127 21 124)(3 128 22 125)(4 129 23 126)(5 130 24 121)(6 131 19 122)(7 140 148 160)(8 141 149 161)(9 142 150 162)(10 143 145 157)(11 144 146 158)(12 139 147 159)(13 134 47 156)(14 135 48 151)(15 136 43 152)(16 137 44 153)(17 138 45 154)(18 133 46 155)(25 87 70 103)(26 88 71 104)(27 89 72 105)(28 90 67 106)(29 85 68 107)(30 86 69 108)(31 171 184 180)(32 172 185 175)(33 173 186 176)(34 174 181 177)(35 169 182 178)(36 170 183 179)(37 98 192 163)(38 99 187 164)(39 100 188 165)(40 101 189 166)(41 102 190 167)(42 97 191 168)(49 57 120 61)(50 58 115 62)(51 59 116 63)(52 60 117 64)(53 55 118 65)(54 56 119 66)(73 92 82 112)(74 93 83 113)(75 94 84 114)(76 95 79 109)(77 96 80 110)(78 91 81 111)

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

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

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

120 conjugacy classes

 class 1 2A ··· 2O 3A 3B 4A ··· 4X 6A ··· 6AD 12A ··· 12AV order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C3 C4 C6 C6 C12 D4 Q8 C3×D4 C3×Q8 kernel C6×C2.C42 C3×C2.C42 C23×C12 C2×C2.C42 C22×C12 C2.C42 C23×C4 C22×C4 C22×C6 C22×C6 C23 C23 # reps 1 4 3 2 24 8 6 48 6 2 12 4

Matrix representation of C6×C2.C42 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 10 0 0 0 0 0 10
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 8 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 10 1 0 0 0 3 3
,
 12 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 1 2 0 0 0 0 12

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

C6×C2.C42 in GAP, Magma, Sage, TeX

C_6\times C_2.C_4^2
% in TeX

G:=Group("C6xC2.C4^2");
// GroupNames label

G:=SmallGroup(192,808);
// by ID

G=gap.SmallGroup(192,808);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680]);
// Polycyclic

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

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