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## G = S3×C2×C26order 312 = 23·3·13

### Direct product of C2×C26 and S3

Aliases: S3×C2×C26, C394C23, C784C22, C6⋊(C2×C26), C3⋊(C22×C26), (C2×C6)⋊3C26, (C2×C78)⋊7C2, SmallGroup(312,59)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C26
 Chief series C1 — C3 — C39 — S3×C13 — S3×C26 — S3×C2×C26
 Lower central C3 — S3×C2×C26
 Upper central C1 — C2×C26

Generators and relations for S3×C2×C26
G = < a,b,c,d | a2=b26=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 108 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, D6, C2×C6, C13, C22×S3, C26, C26, C39, C2×C26, C2×C26, S3×C13, C78, C22×C26, S3×C26, C2×C78, S3×C2×C26
Quotients: C1, C2, C22, S3, C23, D6, C13, C22×S3, C26, C2×C26, S3×C13, C22×C26, S3×C26, S3×C2×C26

Smallest permutation representation of S3×C2×C26
On 156 points
Generators in S156
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 141)(10 142)(11 143)(12 144)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 151)(20 152)(21 153)(22 154)(23 155)(24 156)(25 131)(26 132)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 53)(45 54)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 105)(102 106)(103 107)(104 108)
(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 91 36)(2 92 37)(3 93 38)(4 94 39)(5 95 40)(6 96 41)(7 97 42)(8 98 43)(9 99 44)(10 100 45)(11 101 46)(12 102 47)(13 103 48)(14 104 49)(15 79 50)(16 80 51)(17 81 52)(18 82 27)(19 83 28)(20 84 29)(21 85 30)(22 86 31)(23 87 32)(24 88 33)(25 89 34)(26 90 35)(53 141 129)(54 142 130)(55 143 105)(56 144 106)(57 145 107)(58 146 108)(59 147 109)(60 148 110)(61 149 111)(62 150 112)(63 151 113)(64 152 114)(65 153 115)(66 154 116)(67 155 117)(68 156 118)(69 131 119)(70 132 120)(71 133 121)(72 134 122)(73 135 123)(74 136 124)(75 137 125)(76 138 126)(77 139 127)(78 140 128)
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 141)(10 142)(11 143)(12 144)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 151)(20 152)(21 153)(22 154)(23 155)(24 156)(25 131)(26 132)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)

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

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

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

156 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C 13A ··· 13L 26A ··· 26AJ 26AK ··· 26CF 39A ··· 39L 78A ··· 78AJ order 1 2 2 2 2 2 2 2 3 6 6 6 13 ··· 13 26 ··· 26 26 ··· 26 39 ··· 39 78 ··· 78 size 1 1 1 1 3 3 3 3 2 2 2 2 1 ··· 1 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

156 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C13 C26 C26 S3 D6 S3×C13 S3×C26 kernel S3×C2×C26 S3×C26 C2×C78 C22×S3 D6 C2×C6 C2×C26 C26 C22 C2 # reps 1 6 1 12 72 12 1 3 12 36

Matrix representation of S3×C2×C26 in GL4(𝔽79) generated by

 78 0 0 0 0 78 0 0 0 0 1 0 0 0 0 1
,
 78 0 0 0 0 1 0 0 0 0 38 0 0 0 0 38
,
 1 0 0 0 0 1 0 0 0 0 78 78 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 78 78
G:=sub<GL(4,GF(79))| [78,0,0,0,0,78,0,0,0,0,1,0,0,0,0,1],[78,0,0,0,0,1,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,1,0,0,0,0,78,1,0,0,78,0],[1,0,0,0,0,1,0,0,0,0,1,78,0,0,0,78] >;

S3×C2×C26 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{26}
% in TeX

G:=Group("S3xC2xC26");
// GroupNames label

G:=SmallGroup(312,59);
// by ID

G=gap.SmallGroup(312,59);
# by ID

G:=PCGroup([5,-2,-2,-2,-13,-3,5204]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^26=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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