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G = C397D4order 312 = 23·3·13

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C397D4, D782C2, C2.5D78, C6.12D26, C26.12D6, C222D39, Dic391C2, C78.12C22, (C2×C26)⋊4S3, (C2×C78)⋊2C2, (C2×C6)⋊2D13, C133(C3⋊D4), C33(C13⋊D4), SmallGroup(312,41)

Series: Derived Chief Lower central Upper central

C1C78 — C397D4
C1C13C39C78D78 — C397D4
C39C78 — C397D4
C1C2C22

Generators and relations for C397D4
 G = < a,b,c | a39=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
78C2
39C22
39C4
2C6
26S3
2C26
6D13
39D4
13D6
13Dic3
3D26
3Dic13
2C78
2D39
13C3⋊D4
3C13⋊D4

Smallest permutation representation of C397D4
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)
(1 106 41 142)(2 105 42 141)(3 104 43 140)(4 103 44 139)(5 102 45 138)(6 101 46 137)(7 100 47 136)(8 99 48 135)(9 98 49 134)(10 97 50 133)(11 96 51 132)(12 95 52 131)(13 94 53 130)(14 93 54 129)(15 92 55 128)(16 91 56 127)(17 90 57 126)(18 89 58 125)(19 88 59 124)(20 87 60 123)(21 86 61 122)(22 85 62 121)(23 84 63 120)(24 83 64 119)(25 82 65 118)(26 81 66 156)(27 80 67 155)(28 79 68 154)(29 117 69 153)(30 116 70 152)(31 115 71 151)(32 114 72 150)(33 113 73 149)(34 112 74 148)(35 111 75 147)(36 110 76 146)(37 109 77 145)(38 108 78 144)(39 107 40 143)
(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(40 42)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(79 130)(80 129)(81 128)(82 127)(83 126)(84 125)(85 124)(86 123)(87 122)(88 121)(89 120)(90 119)(91 118)(92 156)(93 155)(94 154)(95 153)(96 152)(97 151)(98 150)(99 149)(100 148)(101 147)(102 146)(103 145)(104 144)(105 143)(106 142)(107 141)(108 140)(109 139)(110 138)(111 137)(112 136)(113 135)(114 134)(115 133)(116 132)(117 131)

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

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

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

81 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C13A···13F26A···26R39A···39L78A···78AJ
order12223466613···1326···2639···3978···78
size112782782222···22···22···22···2

81 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3D4D6C3⋊D4D13D26D39C13⋊D4D78C397D4
kernelC397D4Dic39D78C2×C78C2×C26C39C26C13C2×C6C6C22C3C2C1
# reps111111126612121224

Matrix representation of C397D4 in GL4(𝔽157) generated by

117500
8213100
0010920
0013777
,
153400
5814200
003197
0016126
,
112400
015600
003197
0016126
G:=sub<GL(4,GF(157))| [11,82,0,0,75,131,0,0,0,0,109,137,0,0,20,77],[15,58,0,0,34,142,0,0,0,0,31,16,0,0,97,126],[1,0,0,0,124,156,0,0,0,0,31,16,0,0,97,126] >;

C397D4 in GAP, Magma, Sage, TeX

C_{39}\rtimes_7D_4
% in TeX

G:=Group("C39:7D4");
// GroupNames label

G:=SmallGroup(312,41);
// by ID

G=gap.SmallGroup(312,41);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-13,61,323,7204]);
// Polycyclic

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

Export

Subgroup lattice of C397D4 in TeX

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