C3:S3xC2^2xC6 - GroupNames

G = C₃:S₃xC₂²xC₆ order 432 = 2⁴·3³

Direct product of C₂²xC₆ and C₃:S₃

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₃:S₃xC₂²xC₆, C₆³:₆C₂, C₆²:₃₅D₆, C₃³:₆C₂⁴, (C₂xC₆²):₁₃S₃, C₆²:₂₁(C₂xC₆), (C₂xC₆²):₁₄C₆, C₃²:₈(S₃xC₂³), C₃²:₅(C₂³xC₆), (C₃²xC₆):₆C₂³, (C₃xC₆²):₁₈C₂², C₆:₂(S₃xC₂xC₆), C₃:₂(S₃xC₂²xC₆), (C₂xC₆):₁₄(S₃xC₆), (C₂²xC₆):₇(C₃xS₃), (C₃xC₆):₅(C₂²xC₆), (C₃xC₆):₈(C₂²xS₃), SmallGroup(432,773)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃:S₃xC₂²xC₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃xC₃:S₃ — C₆xC₃:S₃ — C₂xC₆xC₃:S₃ — C₃:S₃xC₂²xC₆

Lower central

C₃² — C₃:S₃xC₂²xC₆

Upper central

C₁ — C₂²xC₆

Generators and relations for C₃:S₃xC₂²xC₆
G = < a,b,c,d,e,f | a²=b²=c⁶=d³=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d^-1, fef=e^-1 >

Subgroups: 2692 in 932 conjugacy classes, 294 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₂², C₂², S₃, C₆, C₆, C₂³, C₂³, C₃², C₃², C₃², D₆, C₂xC₆, C₂xC₆, C₂⁴, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂²xC₆, C₃³, S₃xC₆, C₂xC₃:S₃, C₆², C₆², S₃xC₂³, C₂³xC₆, C₃xC₃:S₃, C₃²xC₆, S₃xC₂xC₆, C₂²xC₃:S₃, C₂xC₆², C₂xC₆², C₂xC₆², C₆xC₃:S₃, C₃xC₆², S₃xC₂²xC₆, C₂³xC₃:S₃, C₂xC₆xC₃:S₃, C₆³, C₃:S₃xC₂²xC₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂xC₆, C₂⁴, C₃xS₃, C₃:S₃, C₂²xS₃, C₂²xC₆, S₃xC₆, C₂xC₃:S₃, S₃xC₂³, C₂³xC₆, C₃xC₃:S₃, S₃xC₂xC₆, C₂²xC₃:S₃, C₆xC₃:S₃, S₃xC₂²xC₆, C₂³xC₃:S₃, C₂xC₆xC₃:S₃, C₃:S₃xC₂²xC₆

Smallest permutation representation of C₃:S₃xC₂²xC₆
►On 144 points

Generators in S₁₄₄

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

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

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

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

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

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

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

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

144 conjugacy classes

class	1	2A	···	2G	2H	···	2O	3A	3B	3C	···	3N	6A	···	6N	6O	···	6CT	6CU	···	6DJ
order	1	2	···	2	2	···	2	3	3	3	···	3	6	···	6	6	···	6	6	···	6
size	1	1	···	1	9	···	9	1	1	2	···	2	1	···	1	2	···	2	9	···	9

144 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2
type	+	+	+				+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃xS₃	S₃xC₆
kernel	C₃:S₃xC₂²xC₆	C₂xC₆xC₃:S₃	C₆³	C₂³xC₃:S₃	C₂²xC₃:S₃	C₂xC₆²	C₂xC₆²	C₆²	C₂²xC₆	C₂xC₆
# reps	1	14	1	2	28	2	4	28	8	56

Matrix representation of C₃:S₃xC₂²xC₆ ►in GL₅(F₇)

6	0	0	0	0
0	6	0	0	0
0	0	6	0	0
0	0	0	6	0
0	0	0	0	6

6	0	0	0	0
0	6	0	0	0
0	0	6	0	0
0	0	0	1	0
0	0	0	0	1

2	0	0	0	0
0	3	0	0	0
0	0	3	0	0
0	0	0	4	0
0	0	0	0	4

1	0	0	0	0
0	2	0	0	0
0	0	4	0	0
0	0	0	4	0
0	0	0	0	2

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	4	0
0	0	0	0	2

1	0	0	0	0
0	0	2	0	0
0	4	0	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(7))| [6,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[6,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,2],[1,0,0,0,0,0,0,4,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₃:S₃xC₂²xC₆ in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_2^2\times C_6

% in TeX

G:=Group("C3:S3xC2^2xC6");

// GroupNames label

G:=SmallGroup(432,773);

// by ID

G=gap.SmallGroup(432,773);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,4037,14118]);

// Polycyclic

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

// generators/relations

G = C3:S3xC22xC6 order 432 = 24·33

Direct product of C22xC6 and C3:S3

G = C₃:S₃xC₂²xC₆ order 432 = 2⁴·3³

Direct product of C₂²xC₆ and C₃:S₃