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## G = C2×C6×C13⋊C3order 468 = 22·32·13

### Direct product of C2×C6 and C13⋊C3

Aliases: C2×C6×C13⋊C3, C784C6, C132C62, (C2×C78)⋊3C3, C262(C3×C6), C396(C2×C6), (C2×C26)⋊4C32, SmallGroup(468,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C2×C6×C13⋊C3
 Chief series C1 — C13 — C39 — C3×C13⋊C3 — C6×C13⋊C3 — C2×C6×C13⋊C3
 Lower central C13 — C2×C6×C13⋊C3
 Upper central C1 — C2×C6

Generators and relations for C2×C6×C13⋊C3
G = < a,b,c,d | a2=b6=c13=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >

Subgroups: 300 in 60 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C3, C3, C22, C6, C6, C32, C2×C6, C2×C6, C13, C3×C6, C26, C62, C13⋊C3, C39, C2×C26, C2×C13⋊C3, C78, C3×C13⋊C3, C22×C13⋊C3, C2×C78, C6×C13⋊C3, C2×C6×C13⋊C3
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, C62, C13⋊C3, C2×C13⋊C3, C3×C13⋊C3, C22×C13⋊C3, C6×C13⋊C3, C2×C6×C13⋊C3

Smallest permutation representation of C2×C6×C13⋊C3
On 156 points
Generators in S156
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)
(1 53 27 40 14 66)(2 54 28 41 15 67)(3 55 29 42 16 68)(4 56 30 43 17 69)(5 57 31 44 18 70)(6 58 32 45 19 71)(7 59 33 46 20 72)(8 60 34 47 21 73)(9 61 35 48 22 74)(10 62 36 49 23 75)(11 63 37 50 24 76)(12 64 38 51 25 77)(13 65 39 52 26 78)(79 131 105 118 92 144)(80 132 106 119 93 145)(81 133 107 120 94 146)(82 134 108 121 95 147)(83 135 109 122 96 148)(84 136 110 123 97 149)(85 137 111 124 98 150)(86 138 112 125 99 151)(87 139 113 126 100 152)(88 140 114 127 101 153)(89 141 115 128 102 154)(90 142 116 129 103 155)(91 143 117 130 104 156)
(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)
(1 14 27)(2 17 36)(3 20 32)(4 23 28)(5 26 37)(6 16 33)(7 19 29)(8 22 38)(9 25 34)(10 15 30)(11 18 39)(12 21 35)(13 24 31)(40 53 66)(41 56 75)(42 59 71)(43 62 67)(44 65 76)(45 55 72)(46 58 68)(47 61 77)(48 64 73)(49 54 69)(50 57 78)(51 60 74)(52 63 70)(79 92 105)(80 95 114)(81 98 110)(82 101 106)(83 104 115)(84 94 111)(85 97 107)(86 100 116)(87 103 112)(88 93 108)(89 96 117)(90 99 113)(91 102 109)(118 131 144)(119 134 153)(120 137 149)(121 140 145)(122 143 154)(123 133 150)(124 136 146)(125 139 155)(126 142 151)(127 132 147)(128 135 156)(129 138 152)(130 141 148)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 6A ··· 6F 6G ··· 6X 13A 13B 13C 13D 26A ··· 26L 39A ··· 39H 78A ··· 78X order 1 2 2 2 3 3 3 ··· 3 6 ··· 6 6 ··· 6 13 13 13 13 26 ··· 26 39 ··· 39 78 ··· 78 size 1 1 1 1 1 1 13 ··· 13 1 ··· 1 13 ··· 13 3 3 3 3 3 ··· 3 3 ··· 3 3 ··· 3

84 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 C13⋊C3 C2×C13⋊C3 C3×C13⋊C3 C6×C13⋊C3 kernel C2×C6×C13⋊C3 C6×C13⋊C3 C22×C13⋊C3 C2×C78 C2×C13⋊C3 C78 C2×C6 C6 C22 C2 # reps 1 3 6 2 18 6 4 12 8 24

Matrix representation of C2×C6×C13⋊C3 in GL4(𝔽79) generated by

 78 0 0 0 0 78 0 0 0 0 78 0 0 0 0 78
,
 23 0 0 0 0 56 0 0 0 0 56 0 0 0 0 56
,
 1 0 0 0 0 3 55 35 0 1 0 39 0 0 1 66
,
 1 0 0 0 0 17 3 2 0 70 8 52 0 4 73 54
G:=sub<GL(4,GF(79))| [78,0,0,0,0,78,0,0,0,0,78,0,0,0,0,78],[23,0,0,0,0,56,0,0,0,0,56,0,0,0,0,56],[1,0,0,0,0,3,1,0,0,55,0,1,0,35,39,66],[1,0,0,0,0,17,70,4,0,3,8,73,0,2,52,54] >;

C2×C6×C13⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_{13}\rtimes C_3
% in TeX

G:=Group("C2xC6xC13:C3");
// GroupNames label

G:=SmallGroup(468,47);
// by ID

G=gap.SmallGroup(468,47);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,689]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^13=d^3=1,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^9>;
// generators/relations

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