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G = C13×C3⋊D4order 312 = 23·3·13

Direct product of C13 and C3⋊D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C13×C3⋊D4, C399D4, D62C26, Dic3⋊C26, C26.17D6, C78.22C22, C32(D4×C13), (C2×C26)⋊3S3, (C2×C6)⋊2C26, (C2×C78)⋊6C2, (S3×C26)⋊5C2, C2.5(S3×C26), C6.5(C2×C26), C222(S3×C13), (Dic3×C13)⋊4C2, SmallGroup(312,36)

Series: Derived Chief Lower central Upper central

C1C6 — C13×C3⋊D4
C1C3C6C78S3×C26 — C13×C3⋊D4
C3C6 — C13×C3⋊D4
C1C26C2×C26

Generators and relations for C13×C3⋊D4
 G = < a,b,c,d | a13=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

2C2
6C2
3C22
3C4
2C6
2S3
2C26
6C26
3D4
3C2×C26
3C52
2C78
2S3×C13
3D4×C13

Smallest permutation representation of C13×C3⋊D4
On 156 points
Generators in S156
(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 80 32)(2 81 33)(3 82 34)(4 83 35)(5 84 36)(6 85 37)(7 86 38)(8 87 39)(9 88 27)(10 89 28)(11 90 29)(12 91 30)(13 79 31)(14 52 125)(15 40 126)(16 41 127)(17 42 128)(18 43 129)(19 44 130)(20 45 118)(21 46 119)(22 47 120)(23 48 121)(24 49 122)(25 50 123)(26 51 124)(53 72 135)(54 73 136)(55 74 137)(56 75 138)(57 76 139)(58 77 140)(59 78 141)(60 66 142)(61 67 143)(62 68 131)(63 69 132)(64 70 133)(65 71 134)(92 148 117)(93 149 105)(94 150 106)(95 151 107)(96 152 108)(97 153 109)(98 154 110)(99 155 111)(100 156 112)(101 144 113)(102 145 114)(103 146 115)(104 147 116)
(1 156 135 44)(2 144 136 45)(3 145 137 46)(4 146 138 47)(5 147 139 48)(6 148 140 49)(7 149 141 50)(8 150 142 51)(9 151 143 52)(10 152 131 40)(11 153 132 41)(12 154 133 42)(13 155 134 43)(14 88 95 61)(15 89 96 62)(16 90 97 63)(17 91 98 64)(18 79 99 65)(19 80 100 53)(20 81 101 54)(21 82 102 55)(22 83 103 56)(23 84 104 57)(24 85 92 58)(25 86 93 59)(26 87 94 60)(27 107 67 125)(28 108 68 126)(29 109 69 127)(30 110 70 128)(31 111 71 129)(32 112 72 130)(33 113 73 118)(34 114 74 119)(35 115 75 120)(36 116 76 121)(37 117 77 122)(38 105 78 123)(39 106 66 124)
(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 105)(26 106)(27 88)(28 89)(29 90)(30 91)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 152)(41 153)(42 154)(43 155)(44 156)(45 144)(46 145)(47 146)(48 147)(49 148)(50 149)(51 150)(52 151)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 66)(61 67)(62 68)(63 69)(64 70)(65 71)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 118)(102 119)(103 120)(104 121)

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

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

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

117 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C13A···13L26A···26L26M···26X26Y···26AJ39A···39L52A···52L78A···78AJ
order12223466613···1326···2626···2626···2639···3952···5278···78
size1126262221···11···12···26···62···26···62···2

117 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C13C26C26C26S3D4D6C3⋊D4S3×C13D4×C13S3×C26C13×C3⋊D4
kernelC13×C3⋊D4Dic3×C13S3×C26C2×C78C3⋊D4Dic3D6C2×C6C2×C26C39C26C13C22C3C2C1
# reps111112121212111212121224

Matrix representation of C13×C3⋊D4 in GL2(𝔽157) generated by

1300
0130
,
156156
10
,
4488
44113
,
10
156156
G:=sub<GL(2,GF(157))| [130,0,0,130],[156,1,156,0],[44,44,88,113],[1,156,0,156] >;

C13×C3⋊D4 in GAP, Magma, Sage, TeX

C_{13}\times C_3\rtimes D_4
% in TeX

G:=Group("C13xC3:D4");
// GroupNames label

G:=SmallGroup(312,36);
// by ID

G=gap.SmallGroup(312,36);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-3,541,5204]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of C13×C3⋊D4 in TeX

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