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G = D4×C39order 312 = 23·3·13

Direct product of C39 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C39, C4⋊C78, C527C6, C123C26, C1567C2, C222C78, C78.23C22, (C2×C6)⋊1C26, (C2×C78)⋊1C2, (C2×C26)⋊9C6, C2.1(C2×C78), C6.6(C2×C26), C26.14(C2×C6), SmallGroup(312,43)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C39
C1C2C26C78C2×C78 — D4×C39
C1C2 — D4×C39
C1C78 — D4×C39

Generators and relations for D4×C39
 G = < a,b,c | a39=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C6
2C6
2C26
2C26
2C78
2C78

Smallest permutation representation of D4×C39
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)
(1 100 122 70)(2 101 123 71)(3 102 124 72)(4 103 125 73)(5 104 126 74)(6 105 127 75)(7 106 128 76)(8 107 129 77)(9 108 130 78)(10 109 131 40)(11 110 132 41)(12 111 133 42)(13 112 134 43)(14 113 135 44)(15 114 136 45)(16 115 137 46)(17 116 138 47)(18 117 139 48)(19 79 140 49)(20 80 141 50)(21 81 142 51)(22 82 143 52)(23 83 144 53)(24 84 145 54)(25 85 146 55)(26 86 147 56)(27 87 148 57)(28 88 149 58)(29 89 150 59)(30 90 151 60)(31 91 152 61)(32 92 153 62)(33 93 154 63)(34 94 155 64)(35 95 156 65)(36 96 118 66)(37 97 119 67)(38 98 120 68)(39 99 121 69)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(79 140)(80 141)(81 142)(82 143)(83 144)(84 145)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)

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

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

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

195 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F12A12B13A···13L26A···26L26M···26AJ39A···39X52A···52L78A···78X78Y···78BT156A···156X
order1222334666666121213···1326···2626···2639···3952···5278···7878···78156···156
size1122112112222221···11···12···21···12···21···12···22···2

195 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C3C6C6C13C26C26C39C78C78D4C3×D4D4×C13D4×C39
kernelD4×C39C156C2×C78D4×C13C52C2×C26C3×D4C12C2×C6D4C4C22C39C13C3C1
# reps112224121224242448121224

Matrix representation of D4×C39 in GL2(𝔽157) generated by

1320
0132
,
109155
13248
,
109155
13148
G:=sub<GL(2,GF(157))| [132,0,0,132],[109,132,155,48],[109,131,155,48] >;

D4×C39 in GAP, Magma, Sage, TeX

D_4\times C_{39}
% in TeX

G:=Group("D4xC39");
// GroupNames label

G:=SmallGroup(312,43);
// by ID

G=gap.SmallGroup(312,43);
# by ID

G:=PCGroup([5,-2,-2,-3,-13,-2,1581]);
// Polycyclic

G:=Group<a,b,c|a^39=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D4×C39 in TeX

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