Copied to
clipboard

## G = (C2×C4).Dic6order 192 = 26·3

### 9th non-split extension by C2×C4 of Dic6 acting via Dic6/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C4).Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×Dic3⋊C4 — (C2×C4).Dic6
 Lower central C3 — C22×C6 — (C2×C4).Dic6
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C4).Dic6
G = < a,b,c,d | a2=b4=c12=1, d2=b2c6, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 336 in 134 conjugacy classes, 57 normal (51 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, C22×Dic3, C22×C12, C23.83C23, C6.C42, C3×C2.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, (C2×C4).Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×Dic6, C4○D12, S3×D4, D42S3, Q83S3, C23.83C23, C12.6Q8, Dic3.D4, C23.8D6, C23.11D6, C4.Dic6, D6.D4, C4⋊C4⋊S3, (C2×C4).Dic6

Smallest permutation representation of (C2×C4).Dic6
Regular action on 192 points
Generators in S192
(1 155)(2 156)(3 145)(4 146)(5 147)(6 148)(7 149)(8 150)(9 151)(10 152)(11 153)(12 154)(13 186)(14 187)(15 188)(16 189)(17 190)(18 191)(19 192)(20 181)(21 182)(22 183)(23 184)(24 185)(25 44)(26 45)(27 46)(28 47)(29 48)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)(49 68)(50 69)(51 70)(52 71)(53 72)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 109)(105 110)(106 111)(107 112)(108 113)(121 140)(122 141)(123 142)(124 143)(125 144)(126 133)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(157 177)(158 178)(159 179)(160 180)(161 169)(162 170)(163 171)(164 172)(165 173)(166 174)(167 175)(168 176)
(1 97 83 27)(2 115 84 47)(3 99 73 29)(4 117 74 37)(5 101 75 31)(6 119 76 39)(7 103 77 33)(8 109 78 41)(9 105 79 35)(10 111 80 43)(11 107 81 25)(12 113 82 45)(13 53 164 137)(14 61 165 131)(15 55 166 139)(16 63 167 121)(17 57 168 141)(18 65 157 123)(19 59 158 143)(20 67 159 125)(21 49 160 133)(22 69 161 127)(23 51 162 135)(24 71 163 129)(26 154 108 88)(28 156 98 90)(30 146 100 92)(32 148 102 94)(34 150 104 96)(36 152 106 86)(38 147 118 93)(40 149 120 95)(42 151 110 85)(44 153 112 87)(46 155 114 89)(48 145 116 91)(50 169 134 183)(52 171 136 185)(54 173 138 187)(56 175 140 189)(58 177 142 191)(60 179 144 181)(62 174 132 188)(64 176 122 190)(66 178 124 192)(68 180 126 182)(70 170 128 184)(72 172 130 186)
(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)
(1 181 77 173)(2 178 78 186)(3 191 79 171)(4 176 80 184)(5 189 81 169)(6 174 82 182)(7 187 83 179)(8 172 84 192)(9 185 73 177)(10 170 74 190)(11 183 75 175)(12 180 76 188)(13 156 158 96)(14 89 159 149)(15 154 160 94)(16 87 161 147)(17 152 162 92)(18 85 163 145)(19 150 164 90)(20 95 165 155)(21 148 166 88)(22 93 167 153)(23 146 168 86)(24 91 157 151)(25 50 101 140)(26 133 102 55)(27 60 103 138)(28 143 104 53)(29 58 105 136)(30 141 106 51)(31 56 107 134)(32 139 108 49)(33 54 97 144)(34 137 98 59)(35 52 99 142)(36 135 100 57)(37 122 111 70)(38 63 112 127)(39 132 113 68)(40 61 114 125)(41 130 115 66)(42 71 116 123)(43 128 117 64)(44 69 118 121)(45 126 119 62)(46 67 120 131)(47 124 109 72)(48 65 110 129)

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4F 4G ··· 4N 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + - + - + image C1 C2 C2 C2 C2 S3 D4 Q8 D6 C4○D4 Dic6 C4○D12 S3×D4 D4⋊2S3 Q8⋊3S3 kernel (C2×C4).Dic6 C6.C42 C3×C2.C42 C2×Dic3⋊C4 C2×C4⋊Dic3 C2.C42 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C22 C22 C22 C22 # reps 1 4 1 1 1 1 2 2 3 10 4 8 1 2 1

Matrix representation of (C2×C4).Dic6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 8 0 0 0 0 8 0
,
 6 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 3 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 0 2 0 0 0 0 7 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[6,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[0,7,0,0,0,0,2,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;

(C2×C4).Dic6 in GAP, Magma, Sage, TeX

(C_2\times C_4).{\rm Dic}_6
% in TeX

G:=Group("(C2xC4).Dic6");
// GroupNames label

G:=SmallGroup(192,219);
// by ID

G=gap.SmallGroup(192,219);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,64,254,387,268,6278]);
// Polycyclic

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

׿
×
𝔽