C2^2xC4xHe3 - GroupNames

G = C₂²xC₄xHe₃ order 432 = 2⁴·3³

Direct product of C₂²xC₄ and He₃

direct product, metabelian, nilpotent (class 2), monomial

Aliases: C₂²xC₄xHe₃, C₆²:₁₁C₁₂, C₁₂.₄₃C₆², (C₆xC₁₂):₈C₆, C₆.₁₇(C₆xC₁₂), (C₂xC₆²).₁₄C₆, C₆.₁₁(C₂xC₆²), C₆².₃₇(C₂xC₆), (C₂xC₆).₃₂C₆², C₂.₁(C₂³xHe₃), C₂³.₃(C₂xHe₃), C₃²:₇(C₂²xC₁₂), (C₂³xHe₃).₆C₂, (C₂xHe₃).₃₉C₂³, (C₂²xC₁₂).₆C₃², C₂².₅(C₂²xHe₃), (C₂²xHe₃).₃₉C₂², (C₂xC₆xC₁₂):₂C₃, C₃.₁(C₂xC₆xC₁₂), (C₃xC₆):₆(C₂xC₁₂), (C₃xC₁₂):₁₀(C₂xC₆), (C₂xC₆).₁₇(C₃xC₁₂), (C₂xC₁₂).₃₂(C₃xC₆), (C₂²xC₆).₂₂(C₃xC₆), (C₃xC₆).₂₉(C₂²xC₆), SmallGroup(432,401)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₂²xC₄xHe₃

Chief series

C₁ — C₃ — C₆ — C₃xC₆ — C₂xHe₃ — C₄xHe₃ — C₂xC₄xHe₃ — C₂²xC₄xHe₃

Lower central

C₁ — C₃ — C₂²xC₄xHe₃

Upper central

C₁ — C₂²xC₁₂ — C₂²xC₄xHe₃

Generators and relations for C₂²xC₄xHe₃
G = < a,b,c,d,e,f | a²=b²=c⁴=d³=e³=f³=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^-1=de^-1, ef=fe >

Subgroups: 513 in 297 conjugacy classes, 189 normal (12 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₆, C₂xC₄, C₂³, C₃², C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₂²xC₄, C₃xC₆, C₂xC₁₂, C₂xC₁₂, C₂²xC₆, C₂²xC₆, He₃, C₃xC₁₂, C₆², C₂²xC₁₂, C₂²xC₁₂, C₂xHe₃, C₂xHe₃, C₆xC₁₂, C₂xC₆², C₄xHe₃, C₂²xHe₃, C₂xC₆xC₁₂, C₂xC₄xHe₃, C₂³xHe₃, C₂²xC₄xHe₃
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂xC₄, C₂³, C₃², C₁₂, C₂xC₆, C₂²xC₄, C₃xC₆, C₂xC₁₂, C₂²xC₆, He₃, C₃xC₁₂, C₆², C₂²xC₁₂, C₂xHe₃, C₆xC₁₂, C₂xC₆², C₄xHe₃, C₂²xHe₃, C₂xC₆xC₁₂, C₂xC₄xHe₃, C₂³xHe₃, C₂²xC₄xHe₃

Smallest permutation representation of C₂²xC₄xHe₃
►On 144 points

Generators in S₁₄₄

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

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

(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)

(5 57 115)(6 58 116)(7 59 113)(8 60 114)(9 96 63)(10 93 64)(11 94 61)(12 95 62)(17 127 69)(18 128 70)(19 125 71)(20 126 72)(21 79 133)(22 80 134)(23 77 135)(24 78 136)(25 131 75)(26 132 76)(27 129 73)(28 130 74)(33 51 44)(34 52 41)(35 49 42)(36 50 43)(53 123 101)(54 124 102)(55 121 103)(56 122 104)(85 141 111)(86 142 112)(87 143 109)(88 144 110)

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

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

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

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

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

176 conjugacy classes

class	1	2A	···	2G	3A	3B	3C	···	3J	4A	···	4H	6A	···	6N	6O	···	6BR	12A	···	12P	12Q	···	12CB
order	1	2	···	2	3	3	3	···	3	4	···	4	6	···	6	6	···	6	12	···	12	12	···	12
size	1	1	···	1	1	1	3	···	3	1	···	1	1	···	1	3	···	3	1	···	1	3	···	3

176 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	3
type	+	+	+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	He₃	C₂xHe₃	C₂xHe₃	C₄xHe₃
kernel	C₂²xC₄xHe₃	C₂xC₄xHe₃	C₂³xHe₃	C₂xC₆xC₁₂	C₂²xHe₃	C₆xC₁₂	C₂xC₆²	C₆²	C₂²xC₄	C₂xC₄	C₂³	C₂²
# reps	1	6	1	8	8	48	8	64	2	12	2	16

Matrix representation of C₂²xC₄xHe₃ ►in GL₅(F₁₃)

12	0	0	0	0
0	12	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

12	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

12	0	0	0	0
0	8	0	0	0
0	0	8	0	0
0	0	0	8	0
0	0	0	0	8

9	0	0	0	0
0	3	0	0	0
0	0	1	0	0
0	0	0	9	0
0	0	0	0	3

1	0	0	0	0
0	1	0	0	0
0	0	9	0	0
0	0	0	9	0
0	0	0	0	9

1	0	0	0	0
0	3	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[9,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C₂²xC₄xHe₃ in GAP, Magma, Sage, TeX

C_2^2\times C_4\times {\rm He}_3

% in TeX

G:=Group("C2^2xC4xHe3");

// GroupNames label

G:=SmallGroup(432,401);

// by ID

G=gap.SmallGroup(432,401);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,760]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=d^3=e^3=f^3=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^-1=d*e^-1,e*f=f*e>;

// generators/relations

G = C22xC4xHe3 order 432 = 24·33

Direct product of C22xC4 and He3

G = C₂²xC₄xHe₃ order 432 = 2⁴·3³

Direct product of C₂²xC₄ and He₃