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## G = C42⋊6Dic3order 192 = 26·3

### 1st semidirect product of C42 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C42⋊6Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — C42⋊6Dic3
 Lower central C3 — C6 — C42⋊6Dic3
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C426Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 344 in 178 conjugacy classes, 103 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C42, C4×Dic3, C4×C12, C22×Dic3, C22×C12, C22×C12, C424C4, C6.C42, C2×C4×Dic3, C2×C4×C12, C426Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, C4○D4, C4×S3, C2×Dic3, C22×S3, C2×C42, C42⋊C2, C4×Dic3, S3×C2×C4, C4○D12, C22×Dic3, C424C4, C422S3, C2×C4×Dic3, C23.26D6, C426Dic3

Smallest permutation representation of C426Dic3
Regular action on 192 points
Generators in S192
(1 71 23 58)(2 72 24 59)(3 67 19 60)(4 68 20 55)(5 69 21 56)(6 70 22 57)(7 77 29 63)(8 78 30 64)(9 73 25 65)(10 74 26 66)(11 75 27 61)(12 76 28 62)(13 146 181 133)(14 147 182 134)(15 148 183 135)(16 149 184 136)(17 150 185 137)(18 145 186 138)(31 92 44 79)(32 93 45 80)(33 94 46 81)(34 95 47 82)(35 96 48 83)(36 91 43 84)(37 98 51 85)(38 99 52 86)(39 100 53 87)(40 101 54 88)(41 102 49 89)(42 97 50 90)(103 165 117 151)(104 166 118 152)(105 167 119 153)(106 168 120 154)(107 163 115 155)(108 164 116 156)(109 170 122 157)(110 171 123 158)(111 172 124 159)(112 173 125 160)(113 174 126 161)(114 169 121 162)(127 189 141 175)(128 190 142 176)(129 191 143 177)(130 192 144 178)(131 187 139 179)(132 188 140 180)
(1 38 12 34)(2 39 7 35)(3 40 8 36)(4 41 9 31)(5 42 10 32)(6 37 11 33)(13 157 190 152)(14 158 191 153)(15 159 192 154)(16 160 187 155)(17 161 188 156)(18 162 189 151)(19 54 30 43)(20 49 25 44)(21 50 26 45)(22 51 27 46)(23 52 28 47)(24 53 29 48)(55 89 65 79)(56 90 66 80)(57 85 61 81)(58 86 62 82)(59 87 63 83)(60 88 64 84)(67 101 78 91)(68 102 73 92)(69 97 74 93)(70 98 75 94)(71 99 76 95)(72 100 77 96)(103 145 114 141)(104 146 109 142)(105 147 110 143)(106 148 111 144)(107 149 112 139)(108 150 113 140)(115 136 125 131)(116 137 126 132)(117 138 121 127)(118 133 122 128)(119 134 123 129)(120 135 124 130)(163 184 173 179)(164 185 174 180)(165 186 169 175)(166 181 170 176)(167 182 171 177)(168 183 172 178)
(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)
(1 104 4 107)(2 103 5 106)(3 108 6 105)(7 114 10 111)(8 113 11 110)(9 112 12 109)(13 102 16 99)(14 101 17 98)(15 100 18 97)(19 116 22 119)(20 115 23 118)(21 120 24 117)(25 125 28 122)(26 124 29 121)(27 123 30 126)(31 131 34 128)(32 130 35 127)(33 129 36 132)(37 134 40 137)(38 133 41 136)(39 138 42 135)(43 140 46 143)(44 139 47 142)(45 144 48 141)(49 149 52 146)(50 148 53 145)(51 147 54 150)(55 155 58 152)(56 154 59 151)(57 153 60 156)(61 158 64 161)(62 157 65 160)(63 162 66 159)(67 164 70 167)(68 163 71 166)(69 168 72 165)(73 173 76 170)(74 172 77 169)(75 171 78 174)(79 179 82 176)(80 178 83 175)(81 177 84 180)(85 182 88 185)(86 181 89 184)(87 186 90 183)(91 188 94 191)(92 187 95 190)(93 192 96 189)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C4 C4 S3 Dic3 D6 C4○D4 C4×S3 C4○D12 kernel C42⋊6Dic3 C6.C42 C2×C4×Dic3 C2×C4×C12 C4×Dic3 C4×C12 C2×C42 C42 C22×C4 C2×C6 C2×C4 C22 # reps 1 4 2 1 16 8 1 4 3 8 8 16

Matrix representation of C426Dic3 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 8 0 0 0 0 8
,
 8 0 0 0 0 12 0 0 0 0 5 0 0 0 12 8
,
 12 0 0 0 0 12 0 0 0 0 10 0 0 0 2 4
,
 8 0 0 0 0 8 0 0 0 0 8 2 0 0 0 5
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,12,0,0,0,0,5,12,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,10,2,0,0,0,4],[8,0,0,0,0,8,0,0,0,0,8,0,0,0,2,5] >;

C426Dic3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_6{\rm Dic}_3
% in TeX

G:=Group("C4^2:6Dic3");
// GroupNames label

G:=SmallGroup(192,491);
// by ID

G=gap.SmallGroup(192,491);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,758,100,6278]);
// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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