Copied to
clipboard

G = C4×D39order 312 = 23·3·13

Direct product of C4 and D39

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

Aliases: C4×D39, C522S3, C1562C2, C122D13, C2.1D78, C6.9D26, C26.9D6, D78.2C2, Dic395C2, C78.9C22, C133(C4×S3), C397(C2×C4), C32(C4×D13), SmallGroup(312,38)

Series: Derived Chief Lower central Upper central

C1C39 — C4×D39
C1C13C39C78D78 — C4×D39
C39 — C4×D39
C1C4

Generators and relations for C4×D39
 G = < a,b,c | a4=b39=c2=1, ab=ba, ac=ca, cbc=b-1 >

39C2
39C2
39C4
39C22
13S3
13S3
3D13
3D13
39C2×C4
13Dic3
13D6
3D26
3Dic13
13C4×S3
3C4×D13

Smallest permutation representation of C4×D39
On 156 points
Generators in S156
(1 124 58 109)(2 125 59 110)(3 126 60 111)(4 127 61 112)(5 128 62 113)(6 129 63 114)(7 130 64 115)(8 131 65 116)(9 132 66 117)(10 133 67 79)(11 134 68 80)(12 135 69 81)(13 136 70 82)(14 137 71 83)(15 138 72 84)(16 139 73 85)(17 140 74 86)(18 141 75 87)(19 142 76 88)(20 143 77 89)(21 144 78 90)(22 145 40 91)(23 146 41 92)(24 147 42 93)(25 148 43 94)(26 149 44 95)(27 150 45 96)(28 151 46 97)(29 152 47 98)(30 153 48 99)(31 154 49 100)(32 155 50 101)(33 156 51 102)(34 118 52 103)(35 119 53 104)(36 120 54 105)(37 121 55 106)(38 122 56 107)(39 123 57 108)
(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 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 78)(20 77)(21 76)(22 75)(23 74)(24 73)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(79 153)(80 152)(81 151)(82 150)(83 149)(84 148)(85 147)(86 146)(87 145)(88 144)(89 143)(90 142)(91 141)(92 140)(93 139)(94 138)(95 137)(96 136)(97 135)(98 134)(99 133)(100 132)(101 131)(102 130)(103 129)(104 128)(105 127)(106 126)(107 125)(108 124)(109 123)(110 122)(111 121)(112 120)(113 119)(114 118)(115 156)(116 155)(117 154)

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 12A12B13A···13F26A···26F39A···39L52A···52L78A···78L156A···156X
order1222344446121213···1326···2639···3952···5278···78156···156
size11393921139392222···22···22···22···22···22···2

84 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4S3D6C4×S3D13D26D39C4×D13D78C4×D39
kernelC4×D39Dic39C156D78D39C52C26C13C12C6C4C3C2C1
# reps111141126612121224

Matrix representation of C4×D39 in GL2(𝔽157) generated by

280
028
,
6962
9574
,
74132
6283
G:=sub<GL(2,GF(157))| [28,0,0,28],[69,95,62,74],[74,62,132,83] >;

C4×D39 in GAP, Magma, Sage, TeX

C_4\times D_{39}
% in TeX

G:=Group("C4xD39");
// GroupNames label

G:=SmallGroup(312,38);
// by ID

G=gap.SmallGroup(312,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-13,26,323,7204]);
// Polycyclic

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

Export

Subgroup lattice of C4×D39 in TeX

׿
×
𝔽