Copied to
clipboard

## G = Dic3×C42order 192 = 26·3

### Direct product of C42 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C42
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — Dic3×C42
 Lower central C3 — Dic3×C42
 Upper central C1 — C2×C42

Generators and relations for Dic3×C42
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 408 in 258 conjugacy classes, 183 normal (9 characteristic)
C1, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C42, C22×C4, C22×C4, C2×Dic3, C2×C12, C22×C6, C2×C42, C2×C42, C4×Dic3, C4×C12, C22×Dic3, C22×C12, C43, C2×C4×Dic3, C2×C4×C12, Dic3×C42
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, C4×S3, C2×Dic3, C22×S3, C2×C42, C4×Dic3, S3×C2×C4, C22×Dic3, C43, S3×C42, C2×C4×Dic3, Dic3×C42

Smallest permutation representation of Dic3×C42
Regular action on 192 points
Generators in S192
(1 82 34 62)(2 83 35 63)(3 84 36 64)(4 79 31 65)(5 80 32 66)(6 81 33 61)(7 71 17 49)(8 72 18 50)(9 67 13 51)(10 68 14 52)(11 69 15 53)(12 70 16 54)(19 152 181 133)(20 153 182 134)(21 154 183 135)(22 155 184 136)(23 156 185 137)(24 151 186 138)(25 95 47 73)(26 96 48 74)(27 91 43 75)(28 92 44 76)(29 93 45 77)(30 94 46 78)(37 104 57 85)(38 105 58 86)(39 106 59 87)(40 107 60 88)(41 108 55 89)(42 103 56 90)(97 168 120 145)(98 163 115 146)(99 164 116 147)(100 165 117 148)(101 166 118 149)(102 167 119 150)(109 176 128 157)(110 177 129 158)(111 178 130 159)(112 179 131 160)(113 180 132 161)(114 175 127 162)(121 192 144 169)(122 187 139 170)(123 188 140 171)(124 189 141 172)(125 190 142 173)(126 191 143 174)
(1 41 7 28)(2 42 8 29)(3 37 9 30)(4 38 10 25)(5 39 11 26)(6 40 12 27)(13 46 36 57)(14 47 31 58)(15 48 32 59)(16 43 33 60)(17 44 34 55)(18 45 35 56)(19 163 190 179)(20 164 191 180)(21 165 192 175)(22 166 187 176)(23 167 188 177)(24 168 189 178)(49 76 62 89)(50 77 63 90)(51 78 64 85)(52 73 65 86)(53 74 66 87)(54 75 61 88)(67 94 84 104)(68 95 79 105)(69 96 80 106)(70 91 81 107)(71 92 82 108)(72 93 83 103)(97 124 111 138)(98 125 112 133)(99 126 113 134)(100 121 114 135)(101 122 109 136)(102 123 110 137)(115 142 131 152)(116 143 132 153)(117 144 127 154)(118 139 128 155)(119 140 129 156)(120 141 130 151)(145 172 159 186)(146 173 160 181)(147 174 161 182)(148 169 162 183)(149 170 157 184)(150 171 158 185)
(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)
(1 142 4 139)(2 141 5 144)(3 140 6 143)(7 152 10 155)(8 151 11 154)(9 156 12 153)(13 137 16 134)(14 136 17 133)(15 135 18 138)(19 52 22 49)(20 51 23 54)(21 50 24 53)(25 118 28 115)(26 117 29 120)(27 116 30 119)(31 122 34 125)(32 121 35 124)(33 126 36 123)(37 129 40 132)(38 128 41 131)(39 127 42 130)(43 99 46 102)(44 98 47 101)(45 97 48 100)(55 112 58 109)(56 111 59 114)(57 110 60 113)(61 191 64 188)(62 190 65 187)(63 189 66 192)(67 185 70 182)(68 184 71 181)(69 183 72 186)(73 166 76 163)(74 165 77 168)(75 164 78 167)(79 170 82 173)(80 169 83 172)(81 174 84 171)(85 177 88 180)(86 176 89 179)(87 175 90 178)(91 147 94 150)(92 146 95 149)(93 145 96 148)(103 159 106 162)(104 158 107 161)(105 157 108 160)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4X 4Y ··· 4BD 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 S3 Dic3 D6 C4×S3 kernel Dic3×C42 C2×C4×Dic3 C2×C4×C12 C4×Dic3 C4×C12 C2×C42 C42 C22×C4 C2×C4 # reps 1 6 1 48 8 1 4 3 24

Matrix representation of Dic3×C42 in GL4(𝔽13) generated by

 12 0 0 0 0 8 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 12 0 0 0 0 8 0 0 0 0 8
,
 12 0 0 0 0 1 0 0 0 0 0 12 0 0 1 1
,
 8 0 0 0 0 12 0 0 0 0 2 11 0 0 9 11
G:=sub<GL(4,GF(13))| [12,0,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,1,0,0,0,0,0,1,0,0,12,1],[8,0,0,0,0,12,0,0,0,0,2,9,0,0,11,11] >;

Dic3×C42 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_4^2
% in TeX

G:=Group("Dic3xC4^2");
// GroupNames label

G:=SmallGroup(192,489);
// by ID

G=gap.SmallGroup(192,489);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,100,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽