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G = C3×D52order 312 = 23·3·13

Direct product of C3 and D52

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

Aliases: C3×D52, C395D4, C525C6, C1563C2, D264C6, C123D13, C6.15D26, C78.15C22, C4⋊(C3×D13), C134(C3×D4), (C6×D13)⋊4C2, C2.4(C6×D13), C26.11(C2×C6), SmallGroup(312,29)

Series: Derived Chief Lower central Upper central

C1C26 — C3×D52
C1C13C26C78C6×D13 — C3×D52
C13C26 — C3×D52
C1C6C12

Generators and relations for C3×D52
 G = < a,b,c | a3=b52=c2=1, ab=ba, ac=ca, cbc=b-1 >

26C2
26C2
13C22
13C22
26C6
26C6
2D13
2D13
13D4
13C2×C6
13C2×C6
2C3×D13
2C3×D13
13C3×D4

Smallest permutation representation of C3×D52
On 156 points
Generators in S156
(1 117 87)(2 118 88)(3 119 89)(4 120 90)(5 121 91)(6 122 92)(7 123 93)(8 124 94)(9 125 95)(10 126 96)(11 127 97)(12 128 98)(13 129 99)(14 130 100)(15 131 101)(16 132 102)(17 133 103)(18 134 104)(19 135 53)(20 136 54)(21 137 55)(22 138 56)(23 139 57)(24 140 58)(25 141 59)(26 142 60)(27 143 61)(28 144 62)(29 145 63)(30 146 64)(31 147 65)(32 148 66)(33 149 67)(34 150 68)(35 151 69)(36 152 70)(37 153 71)(38 154 72)(39 155 73)(40 156 74)(41 105 75)(42 106 76)(43 107 77)(44 108 78)(45 109 79)(46 110 80)(47 111 81)(48 112 82)(49 113 83)(50 114 84)(51 115 85)(52 116 86)
(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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(69 104)(70 103)(71 102)(72 101)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(105 128)(106 127)(107 126)(108 125)(109 124)(110 123)(111 122)(112 121)(113 120)(114 119)(115 118)(116 117)(129 156)(130 155)(131 154)(132 153)(133 152)(134 151)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)(141 144)(142 143)

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

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

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

87 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F12A12B13A···13F26A···26F39A···39L52A···52L78A···78L156A···156X
order1222334666666121213···1326···2639···3952···5278···78156···156
size1126261121126262626222···22···22···22···22···22···2

87 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6D4C3×D4D13D26C3×D13D52C6×D13C3×D52
kernelC3×D52C156C6×D13D52C52D26C39C13C12C6C4C3C2C1
# reps112224126612121224

Matrix representation of C3×D52 in GL4(𝔽157) generated by

144000
014400
0010
0001
,
1509600
362200
0074116
005220
,
1505100
36700
007652
005581
G:=sub<GL(4,GF(157))| [144,0,0,0,0,144,0,0,0,0,1,0,0,0,0,1],[150,36,0,0,96,22,0,0,0,0,74,52,0,0,116,20],[150,36,0,0,51,7,0,0,0,0,76,55,0,0,52,81] >;

C3×D52 in GAP, Magma, Sage, TeX

C_3\times D_{52}
% in TeX

G:=Group("C3xD52");
// GroupNames label

G:=SmallGroup(312,29);
// by ID

G=gap.SmallGroup(312,29);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,141,66,7204]);
// Polycyclic

G:=Group<a,b,c|a^3=b^52=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D52 in TeX

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