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G = C39:D4order 312 = 23·3·13

1st semidirect product of C39 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C39:1D4, D6:1D13, D26:1S3, C6.4D26, C26.4D6, Dic39:4C2, C78.4C22, (S3xC26):1C2, (C6xD13):1C2, C13:2(C3:D4), C3:2(C13:D4), C2.4(S3xD13), SmallGroup(312,18)

Series: Derived Chief Lower central Upper central

C1C78 — C39:D4
C1C13C39C78C6xD13 — C39:D4
C39C78 — C39:D4
C1C2

Generators and relations for C39:D4
 G = < a,b,c | a39=b4=c2=1, bab-1=a-1, cac=a25, cbc=b-1 >

Subgroups: 228 in 32 conjugacy classes, 14 normal (all characteristic)
Quotients: C1, C2, C22, S3, D4, D6, C3:D4, D13, D26, C13:D4, S3xD13, C39:D4
6C2
26C2
3C22
13C22
39C4
2S3
26C6
2D13
6C26
39D4
13C2xC6
13Dic3
3Dic13
3C2xC26
2S3xC13
2C3xD13
13C3:D4
3C13:D4

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

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

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

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

45 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C13A···13F26A···26F26G···26R39A···39F78A···78F
order12223466613···1326···2626···2639···3978···78
size11626278226262···22···26···64···44···4

45 irreducible representations

dim1111222222244
type++++++++++-
imageC1C2C2C2S3D4D6C3:D4D13D26C13:D4S3xD13C39:D4
kernelC39:D4Dic39C6xD13S3xC26D26C39C26C13D6C6C3C2C1
# reps11111112661266

Matrix representation of C39:D4 in GL4(F157) generated by

5213400
231400
0015593
00811
,
13915400
561800
006643
0014791
,
1000
14515600
0010
0001
G:=sub<GL(4,GF(157))| [52,23,0,0,134,14,0,0,0,0,155,81,0,0,93,1],[139,56,0,0,154,18,0,0,0,0,66,147,0,0,43,91],[1,145,0,0,0,156,0,0,0,0,1,0,0,0,0,1] >;

C39:D4 in GAP, Magma, Sage, TeX

C_{39}\rtimes D_4
% in TeX

G:=Group("C39:D4");
// GroupNames label

G:=SmallGroup(312,18);
// by ID

G=gap.SmallGroup(312,18);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-13,61,168,7204]);
// Polycyclic

G:=Group<a,b,c|a^39=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^25,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C39:D4 in TeX

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