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G = C12×D13order 312 = 23·3·13

Direct product of C12 and D13

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C12×D13, C526C6, C1564C2, D26.4C6, C6.14D26, Dic135C6, C78.14C22, C398(C2×C4), C136(C2×C12), C2.1(C6×D13), C26.10(C2×C6), (C6×D13).4C2, (C3×Dic13)⋊5C2, SmallGroup(312,28)

Series: Derived Chief Lower central Upper central

C1C13 — C12×D13
C1C13C26C78C6×D13 — C12×D13
C13 — C12×D13
C1C12

Generators and relations for C12×D13
 G = < a,b,c | a12=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
13C2
13C4
13C22
13C6
13C6
13C2×C4
13C12
13C2×C6
13C2×C12

Smallest permutation representation of C12×D13
On 156 points
Generators in S156
(1 151 43 83 63 127 16 133 34 99 77 107)(2 152 44 84 64 128 17 134 35 100 78 108)(3 153 45 85 65 129 18 135 36 101 66 109)(4 154 46 86 53 130 19 136 37 102 67 110)(5 155 47 87 54 118 20 137 38 103 68 111)(6 156 48 88 55 119 21 138 39 104 69 112)(7 144 49 89 56 120 22 139 27 92 70 113)(8 145 50 90 57 121 23 140 28 93 71 114)(9 146 51 91 58 122 24 141 29 94 72 115)(10 147 52 79 59 123 25 142 30 95 73 116)(11 148 40 80 60 124 26 143 31 96 74 117)(12 149 41 81 61 125 14 131 32 97 75 105)(13 150 42 82 62 126 15 132 33 98 76 106)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 17)(15 16)(18 26)(19 25)(20 24)(21 23)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(40 45)(41 44)(42 43)(46 52)(47 51)(48 50)(53 59)(54 58)(55 57)(60 65)(61 64)(62 63)(66 74)(67 73)(68 72)(69 71)(75 78)(76 77)(79 86)(80 85)(81 84)(82 83)(87 91)(88 90)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(105 108)(106 107)(109 117)(110 116)(111 115)(112 114)(118 122)(119 121)(123 130)(124 129)(125 128)(126 127)(131 134)(132 133)(135 143)(136 142)(137 141)(138 140)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)

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

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

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

96 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F12A12B12C12D12E12F12G12H13A···13F26A···26F39A···39L52A···52L78A···78L156A···156X
order1222334444666666121212121212121213···1326···2639···3952···5278···78156···156
size1113131111131311131313131111131313132···22···22···22···22···22···2

96 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C3C4C6C6C6C12D13D26C3×D13C4×D13C6×D13C12×D13
kernelC12×D13C3×Dic13C156C6×D13C4×D13C3×D13Dic13C52D26D13C12C6C4C3C2C1
# reps11112422286612121224

Matrix representation of C12×D13 in GL2(𝔽157) generated by

220
022
,
1241
1560
,
01
10
G:=sub<GL(2,GF(157))| [22,0,0,22],[124,156,1,0],[0,1,1,0] >;

C12×D13 in GAP, Magma, Sage, TeX

C_{12}\times D_{13}
% in TeX

G:=Group("C12xD13");
// GroupNames label

G:=SmallGroup(312,28);
// by ID

G=gap.SmallGroup(312,28);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,66,7204]);
// Polycyclic

G:=Group<a,b,c|a^12=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C12×D13 in TeX

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