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## G = C22×C4×He3order 432 = 24·33

### Direct product of C22×C4 and He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C22×C4×He3
 Chief series C1 — C3 — C6 — C3×C6 — C2×He3 — C4×He3 — C2×C4×He3 — C22×C4×He3
 Lower central C1 — C3 — C22×C4×He3
 Upper central C1 — C22×C12 — C22×C4×He3

Generators and relations for C22×C4×He3
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 [×6], C3, C3 [×4], C4 [×4], C22 [×7], C6, C6 [×6], C6 [×28], C2×C4 [×6], C23, C32 [×4], C12 [×4], C12 [×16], C2×C6 [×7], C2×C6 [×28], C22×C4, C3×C6 [×28], C2×C12 [×6], C2×C12 [×24], C22×C6, C22×C6 [×4], He3, C3×C12 [×16], C62 [×28], C22×C12, C22×C12 [×4], C2×He3, C2×He3 [×6], C6×C12 [×24], C2×C62 [×4], C4×He3 [×4], C22×He3 [×7], C2×C6×C12 [×4], C2×C4×He3 [×6], C23×He3, C22×C4×He3
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], C23, C32, C12 [×16], C2×C6 [×28], C22×C4, C3×C6 [×7], C2×C12 [×24], C22×C6 [×4], He3, C3×C12 [×4], C62 [×7], C22×C12 [×4], C2×He3 [×7], C6×C12 [×6], C2×C62, C4×He3 [×4], C22×He3 [×7], C2×C6×C12, C2×C4×He3 [×6], C23×He3, C22×C4×He3

Smallest permutation representation of C22×C4×He3
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 C2×He3 C2×He3 C4×He3 kernel C22×C4×He3 C2×C4×He3 C23×He3 C2×C6×C12 C22×He3 C6×C12 C2×C62 C62 C22×C4 C2×C4 C23 C22 # reps 1 6 1 8 8 48 8 64 2 12 2 16

Matrix representation of C22×C4×He3 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] >;

C22×C4×He3 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

׿
×
𝔽