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G = C3×C13⋊D4order 312 = 23·3·13

Direct product of C3 and C13⋊D4

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

Aliases: C3×C13⋊D4, C398D4, D265C6, C6.17D26, Dic134C6, C78.17C22, C135(C3×D4), (C2×C78)⋊4C2, (C2×C6)⋊1D13, (C2×C26)⋊10C6, (C6×D13)⋊5C2, C2.5(C6×D13), C26.13(C2×C6), C222(C3×D13), (C3×Dic13)⋊4C2, SmallGroup(312,31)

Series: Derived Chief Lower central Upper central

C1C26 — C3×C13⋊D4
C1C13C26C78C6×D13 — C3×C13⋊D4
C13C26 — C3×C13⋊D4
C1C6C2×C6

Generators and relations for C3×C13⋊D4
 G = < a,b,c,d | a3=b13=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

2C2
26C2
13C22
13C4
2C6
26C6
2D13
2C26
13D4
13C2×C6
13C12
2C78
2C3×D13
13C3×D4

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

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

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

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

87 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F12A12B13A···13F26A···26R39A···39L78A···78AJ
order1222334666666121213···1326···2639···3978···78
size1122611261122262626262···22···22···22···2

87 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C3C6C6C6D4C3×D4D13D26C3×D13C13⋊D4C6×D13C3×C13⋊D4
kernelC3×C13⋊D4C3×Dic13C6×D13C2×C78C13⋊D4Dic13D26C2×C26C39C13C2×C6C6C22C3C2C1
# reps11112222126612121224

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

120
012
,
67156
10757
,
13836
5119
,
1075
118147
G:=sub<GL(2,GF(157))| [12,0,0,12],[67,107,156,57],[138,51,36,19],[10,118,75,147] >;

C3×C13⋊D4 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC13:D4");
// GroupNames label

G:=SmallGroup(312,31);
// by ID

G=gap.SmallGroup(312,31);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,141,7204]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^13=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 C3×C13⋊D4 in TeX

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