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

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

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

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

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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