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G = D78.C2order 312 = 23·3·13

The non-split extension by D78 of C2 acting faithfully

metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D78.C2, D392C4, C6.3D26, C26.3D6, Dic132S3, Dic32D13, C78.3C22, C132(C4×S3), C396(C2×C4), C31(C4×D13), C2.3(S3×D13), (C3×Dic13)⋊2C2, (Dic3×C13)⋊2C2, SmallGroup(312,17)

Series: Derived Chief Lower central Upper central

C1C39 — D78.C2
C1C13C39C78C3×Dic13 — D78.C2
C39 — D78.C2
C1C2

Generators and relations for D78.C2
 G = < a,b,c | a78=b2=1, c2=a39, bab=a-1, cac-1=a25, cbc-1=a24b >

39C2
39C2
3C4
13C4
39C22
13S3
13S3
3D13
3D13
39C2×C4
13C12
13D6
3C52
3D26
13C4×S3
3C4×D13

Smallest permutation representation of D78.C2
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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(97 156)(98 155)(99 154)(100 153)(101 152)(102 151)(103 150)(104 149)(105 148)(106 147)(107 146)(108 145)(109 144)(110 143)(111 142)(112 141)(113 140)(114 139)(115 138)(116 137)(117 136)(118 135)(119 134)(120 133)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)
(1 127 40 88)(2 152 41 113)(3 99 42 138)(4 124 43 85)(5 149 44 110)(6 96 45 135)(7 121 46 82)(8 146 47 107)(9 93 48 132)(10 118 49 79)(11 143 50 104)(12 90 51 129)(13 115 52 154)(14 140 53 101)(15 87 54 126)(16 112 55 151)(17 137 56 98)(18 84 57 123)(19 109 58 148)(20 134 59 95)(21 81 60 120)(22 106 61 145)(23 131 62 92)(24 156 63 117)(25 103 64 142)(26 128 65 89)(27 153 66 114)(28 100 67 139)(29 125 68 86)(30 150 69 111)(31 97 70 136)(32 122 71 83)(33 147 72 108)(34 94 73 133)(35 119 74 80)(36 144 75 105)(37 91 76 130)(38 116 77 155)(39 141 78 102)

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

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

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

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 12A12B13A···13F26A···26F39A···39F52A···52L78A···78F
order1222344446121213···1326···2639···3952···5278···78
size1139392331313226262···22···24···46···64···4

48 irreducible representations

dim1111122222244
type++++++++++
imageC1C2C2C2C4S3D6C4×S3D13D26C4×D13S3×D13D78.C2
kernelD78.C2Dic3×C13C3×Dic13D78D39Dic13C26C13Dic3C6C3C2C1
# reps11114112661266

Matrix representation of D78.C2 in GL4(𝔽157) generated by

669100
9313300
00117
0046155
,
115600
1514200
00117
000156
,
11711700
1464000
001560
000156
G:=sub<GL(4,GF(157))| [66,93,0,0,91,133,0,0,0,0,1,46,0,0,17,155],[115,151,0,0,6,42,0,0,0,0,1,0,0,0,17,156],[117,146,0,0,117,40,0,0,0,0,156,0,0,0,0,156] >;

D78.C2 in GAP, Magma, Sage, TeX

D_{78}.C_2
% in TeX

G:=Group("D78.C2");
// GroupNames label

G:=SmallGroup(312,17);
// by ID

G=gap.SmallGroup(312,17);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-13,20,26,168,7204]);
// Polycyclic

G:=Group<a,b,c|a^78=b^2=1,c^2=a^39,b*a*b=a^-1,c*a*c^-1=a^25,c*b*c^-1=a^24*b>;
// generators/relations

Export

Subgroup lattice of D78.C2 in TeX

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