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## G = S3×C2×C62order 432 = 24·33

### Direct product of C2×C62 and S3

Aliases: S3×C2×C62, C635C2, C335C24, C6⋊(C2×C62), (C2×C6)⋊6C62, C3⋊(C22×C62), (C2×C62)⋊13C6, C6220(C2×C6), (C32×C6)⋊5C23, C324(C23×C6), (C3×C62)⋊17C22, (C3×C6)⋊4(C22×C6), (C22×C6)⋊5(C3×C6), SmallGroup(432,772)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C62
 Chief series C1 — C3 — C32 — C33 — S3×C32 — S3×C3×C6 — S3×C62 — S3×C2×C62
 Lower central C3 — S3×C2×C62
 Upper central C1 — C2×C62

Generators and relations for S3×C2×C62
G = < a,b,c,d,e | a2=b6=c6=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1672 in 932 conjugacy classes, 498 normal (10 characteristic)
C1, C2, C2, C3, C3, C3, C22, C22, S3, C6, C6, C23, C23, C32, C32, C32, D6, C2×C6, C2×C6, C24, C3×S3, C3×C6, C3×C6, C22×S3, C22×C6, C22×C6, C22×C6, C33, S3×C6, C62, C62, S3×C23, C23×C6, S3×C32, C32×C6, S3×C2×C6, C2×C62, C2×C62, C2×C62, S3×C3×C6, C3×C62, S3×C22×C6, C22×C62, S3×C62, C63, S3×C2×C62
Quotients: C1, C2, C3, C22, S3, C6, C23, C32, D6, C2×C6, C24, C3×S3, C3×C6, C22×S3, C22×C6, S3×C6, C62, S3×C23, C23×C6, S3×C32, S3×C2×C6, C2×C62, S3×C3×C6, S3×C22×C6, C22×C62, S3×C62, S3×C2×C62

Smallest permutation representation of S3×C2×C62
On 144 points
Generators in S144
(1 53)(2 54)(3 49)(4 50)(5 51)(6 52)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 94)(14 95)(15 96)(16 91)(17 92)(18 93)(19 106)(20 107)(21 108)(22 103)(23 104)(24 105)(25 89)(26 90)(27 85)(28 86)(29 87)(30 88)(31 99)(32 100)(33 101)(34 102)(35 97)(36 98)(37 74)(38 75)(39 76)(40 77)(41 78)(42 73)(43 59)(44 60)(45 55)(46 56)(47 57)(48 58)(61 71)(62 72)(63 67)(64 68)(65 69)(66 70)(109 121)(110 122)(111 123)(112 124)(113 125)(114 126)(115 131)(116 132)(117 127)(118 128)(119 129)(120 130)(133 143)(134 144)(135 139)(136 140)(137 141)(138 142)
(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)
(1 57 31 37 15 66)(2 58 32 38 16 61)(3 59 33 39 17 62)(4 60 34 40 18 63)(5 55 35 41 13 64)(6 56 36 42 14 65)(7 141 124 118 21 27)(8 142 125 119 22 28)(9 143 126 120 23 29)(10 144 121 115 24 30)(11 139 122 116 19 25)(12 140 123 117 20 26)(43 101 76 92 72 49)(44 102 77 93 67 50)(45 97 78 94 68 51)(46 98 73 95 69 52)(47 99 74 96 70 53)(48 100 75 91 71 54)(79 137 112 128 108 85)(80 138 113 129 103 86)(81 133 114 130 104 87)(82 134 109 131 105 88)(83 135 110 132 106 89)(84 136 111 127 107 90)
(1 31 15)(2 32 16)(3 33 17)(4 34 18)(5 35 13)(6 36 14)(7 21 124)(8 22 125)(9 23 126)(10 24 121)(11 19 122)(12 20 123)(25 116 139)(26 117 140)(27 118 141)(28 119 142)(29 120 143)(30 115 144)(37 66 57)(38 61 58)(39 62 59)(40 63 60)(41 64 55)(42 65 56)(43 76 72)(44 77 67)(45 78 68)(46 73 69)(47 74 70)(48 75 71)(49 101 92)(50 102 93)(51 97 94)(52 98 95)(53 99 96)(54 100 91)(79 108 112)(80 103 113)(81 104 114)(82 105 109)(83 106 110)(84 107 111)(85 128 137)(86 129 138)(87 130 133)(88 131 134)(89 132 135)(90 127 136)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 59)(8 60)(9 55)(10 56)(11 57)(12 58)(13 120)(14 115)(15 116)(16 117)(17 118)(18 119)(19 66)(20 61)(21 62)(22 63)(23 64)(24 65)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 122)(38 123)(39 124)(40 125)(41 126)(42 121)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)

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

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

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

216 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A ··· 3H 3I ··· 3Q 6A ··· 6BD 6BE ··· 6DO 6DP ··· 6GA order 1 2 ··· 2 2 ··· 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

216 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 kernel S3×C2×C62 S3×C62 C63 S3×C22×C6 S3×C2×C6 C2×C62 C2×C62 C62 C22×C6 C2×C6 # reps 1 14 1 8 112 8 1 7 8 56

Matrix representation of S3×C2×C62 in GL4(𝔽7) generated by

 1 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 3 0 0 0 0 3 0 0 0 0 2 0 0 0 0 2
,
 4 0 0 0 0 2 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 4
,
 1 0 0 0 0 6 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(7))| [1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[3,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,2,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4],[1,0,0,0,0,6,0,0,0,0,0,1,0,0,1,0] >;

S3×C2×C62 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_6^2
% in TeX

G:=Group("S3xC2xC6^2");
// GroupNames label

G:=SmallGroup(432,772);
// by ID

G=gap.SmallGroup(432,772);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^6=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
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𝔽