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## G = C12×C13⋊C3order 468 = 22·32·13

### Direct product of C12 and C13⋊C3

Aliases: C12×C13⋊C3, C156⋊C3, C52⋊C32, C398C12, C78.8C6, C134(C3×C12), C26.2(C3×C6), C2.(C6×C13⋊C3), C6.4(C2×C13⋊C3), (C6×C13⋊C3).4C2, (C2×C13⋊C3).2C6, SmallGroup(468,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C12×C13⋊C3
 Chief series C1 — C13 — C26 — C78 — C6×C13⋊C3 — C12×C13⋊C3
 Lower central C13 — C12×C13⋊C3
 Upper central C1 — C12

Generators and relations for C12×C13⋊C3
G = < a,b,c | a12=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >

Smallest permutation representation of C12×C13⋊C3
On 156 points
Generators in S156
(1 144 40 79 53 118 14 131 27 92 66 105)(2 145 41 80 54 119 15 132 28 93 67 106)(3 146 42 81 55 120 16 133 29 94 68 107)(4 147 43 82 56 121 17 134 30 95 69 108)(5 148 44 83 57 122 18 135 31 96 70 109)(6 149 45 84 58 123 19 136 32 97 71 110)(7 150 46 85 59 124 20 137 33 98 72 111)(8 151 47 86 60 125 21 138 34 99 73 112)(9 152 48 87 61 126 22 139 35 100 74 113)(10 153 49 88 62 127 23 140 36 101 75 114)(11 154 50 89 63 128 24 141 37 102 76 115)(12 155 51 90 64 129 25 142 38 103 77 116)(13 156 52 91 65 130 26 143 39 104 78 117)
(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 27 53)(2 30 62)(3 33 58)(4 36 54)(5 39 63)(6 29 59)(7 32 55)(8 35 64)(9 38 60)(10 28 56)(11 31 65)(12 34 61)(13 37 57)(14 40 66)(15 43 75)(16 46 71)(17 49 67)(18 52 76)(19 42 72)(20 45 68)(21 48 77)(22 51 73)(23 41 69)(24 44 78)(25 47 74)(26 50 70)(79 105 131)(80 108 140)(81 111 136)(82 114 132)(83 117 141)(84 107 137)(85 110 133)(86 113 142)(87 116 138)(88 106 134)(89 109 143)(90 112 139)(91 115 135)(92 118 144)(93 121 153)(94 124 149)(95 127 145)(96 130 154)(97 120 150)(98 123 146)(99 126 155)(100 129 151)(101 119 147)(102 122 156)(103 125 152)(104 128 148)

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

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

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

84 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 4A 4B 6A 6B 6C ··· 6H 12A 12B 12C 12D 12E ··· 12P 13A 13B 13C 13D 26A 26B 26C 26D 39A ··· 39H 52A ··· 52H 78A ··· 78H 156A ··· 156P order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 12 12 12 12 12 ··· 12 13 13 13 13 26 26 26 26 39 ··· 39 52 ··· 52 78 ··· 78 156 ··· 156 size 1 1 1 1 13 ··· 13 1 1 1 1 13 ··· 13 1 1 1 1 13 ··· 13 3 3 3 3 3 3 3 3 3 ··· 3 3 ··· 3 3 ··· 3 3 ··· 3

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C4 C6 C6 C12 C12 C13⋊C3 C2×C13⋊C3 C3×C13⋊C3 C4×C13⋊C3 C6×C13⋊C3 C12×C13⋊C3 kernel C12×C13⋊C3 C6×C13⋊C3 C4×C13⋊C3 C156 C3×C13⋊C3 C2×C13⋊C3 C78 C13⋊C3 C39 C12 C6 C4 C3 C2 C1 # reps 1 1 6 2 2 6 2 12 4 4 4 8 8 8 16

Matrix representation of C12×C13⋊C3 in GL4(𝔽157) generated by

 107 0 0 0 0 156 0 0 0 0 156 0 0 0 0 156
,
 1 0 0 0 0 0 0 1 0 1 0 22 0 0 1 111
,
 144 0 0 0 0 1 1 22 0 0 22 110 0 0 111 134
G:=sub<GL(4,GF(157))| [107,0,0,0,0,156,0,0,0,0,156,0,0,0,0,156],[1,0,0,0,0,0,1,0,0,0,0,1,0,1,22,111],[144,0,0,0,0,1,0,0,0,1,22,111,0,22,110,134] >;

C12×C13⋊C3 in GAP, Magma, Sage, TeX

C_{12}\times C_{13}\rtimes C_3
% in TeX

G:=Group("C12xC13:C3");
// GroupNames label

G:=SmallGroup(468,22);
// by ID

G=gap.SmallGroup(468,22);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-13,90,1359]);
// Polycyclic

G:=Group<a,b,c|a^12=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

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