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## G = C22×C4×3- 1+2order 432 = 24·33

### Direct product of C22×C4 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C22×C4×3- 1+2
 Chief series C1 — C3 — C6 — C3×C6 — C2×3- 1+2 — C4×3- 1+2 — C2×C4×3- 1+2 — C22×C4×3- 1+2
 Lower central C1 — C3 — C22×C4×3- 1+2
 Upper central C1 — C22×C12 — C22×C4×3- 1+2

Generators and relations for C22×C4×3- 1+2
G = < a,b,c,d,e | a2=b2=c4=d9=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 270 in 216 conjugacy classes, 189 normal (16 characteristic)
C1, C2, C2 [×6], C3, C3, C4 [×4], C22 [×7], C6, C6 [×6], C6 [×7], C2×C4 [×6], C23, C9 [×3], C32, C12 [×4], C12 [×4], C2×C6 [×7], C2×C6 [×7], C22×C4, C18 [×21], C3×C6, C3×C6 [×6], C2×C12 [×6], C2×C12 [×6], C22×C6, C22×C6, 3- 1+2, C36 [×12], C2×C18 [×21], C3×C12 [×4], C62 [×7], C22×C12, C22×C12, C2×3- 1+2, C2×3- 1+2 [×6], C2×C36 [×18], C22×C18 [×3], C6×C12 [×6], C2×C62, C4×3- 1+2 [×4], C22×3- 1+2 [×7], C22×C36 [×3], C2×C6×C12, C2×C4×3- 1+2 [×6], C23×3- 1+2, C22×C4×3- 1+2
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], 3- 1+2, C3×C12 [×4], C62 [×7], C22×C12 [×4], C2×3- 1+2 [×7], C6×C12 [×6], C2×C62, C4×3- 1+2 [×4], C22×3- 1+2 [×7], C2×C6×C12, C2×C4×3- 1+2 [×6], C23×3- 1+2, C22×C4×3- 1+2

Smallest permutation representation of C22×C4×3- 1+2
On 144 points
Generators in S144
(1 63)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 107)(11 108)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 53)(20 54)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 73)(36 74)(37 71)(38 72)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 127)(90 128)(91 125)(92 126)(93 118)(94 119)(95 120)(96 121)(97 122)(98 123)(99 124)(109 143)(110 144)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 133)(17 134)(18 135)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(46 64)(47 65)(48 66)(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)(82 100)(83 101)(84 102)(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)(118 136)(119 137)(120 138)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)
(1 135 27 124)(2 127 19 125)(3 128 20 126)(4 129 21 118)(5 130 22 119)(6 131 23 120)(7 132 24 121)(8 133 25 122)(9 134 26 123)(10 37 143 35)(11 38 144 36)(12 39 136 28)(13 40 137 29)(14 41 138 30)(15 42 139 31)(16 43 140 32)(17 44 141 33)(18 45 142 34)(46 93 57 82)(47 94 58 83)(48 95 59 84)(49 96 60 85)(50 97 61 86)(51 98 62 87)(52 99 63 88)(53 91 55 89)(54 92 56 90)(64 111 75 100)(65 112 76 101)(66 113 77 102)(67 114 78 103)(68 115 79 104)(69 116 80 105)(70 117 81 106)(71 109 73 107)(72 110 74 108)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(37 43 40)(38 41 44)(47 53 50)(48 51 54)(55 61 58)(56 59 62)(65 71 68)(66 69 72)(73 79 76)(74 77 80)(83 89 86)(84 87 90)(91 97 94)(92 95 98)(101 107 104)(102 105 108)(109 115 112)(110 113 116)(119 125 122)(120 123 126)(127 133 130)(128 131 134)(137 143 140)(138 141 144)

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

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

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

176 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 4A ··· 4H 6A ··· 6N 6O ··· 6AB 9A ··· 9F 12A ··· 12P 12Q ··· 12AF 18A ··· 18AP 36A ··· 36AV order 1 2 ··· 2 3 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 ··· 1 1 1 3 3 1 ··· 1 1 ··· 1 3 ··· 3 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

176 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 type + + + image C1 C2 C2 C3 C3 C4 C6 C6 C6 C6 C12 C12 3- 1+2 C2×3- 1+2 C2×3- 1+2 C4×3- 1+2 kernel C22×C4×3- 1+2 C2×C4×3- 1+2 C23×3- 1+2 C22×C36 C2×C6×C12 C22×3- 1+2 C2×C36 C22×C18 C6×C12 C2×C62 C2×C18 C62 C22×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 8 36 6 12 2 48 16 2 12 2 16

Matrix representation of C22×C4×3- 1+2 in GL5(𝔽37)

 36 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 1 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 31 0 0 0 0 0 31 0 0 0 0 0 31
,
 10 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 27 27 16 0 0 36 36 10
,
 26 0 0 0 0 0 26 0 0 0 0 0 1 0 0 0 0 0 10 0 0 0 26 27 26

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31],[10,0,0,0,0,0,1,0,0,0,0,0,0,27,36,0,0,1,27,36,0,0,0,16,10],[26,0,0,0,0,0,26,0,0,0,0,0,1,0,26,0,0,0,10,27,0,0,0,0,26] >;

C22×C4×3- 1+2 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times 3_-^{1+2}
% in TeX

G:=Group("C2^2xC4xES-(3,1)");
// GroupNames label

G:=SmallGroup(432,402);
// by ID

G=gap.SmallGroup(432,402);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,528,760]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^9=e^3=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^-1=d^4>;
// generators/relations

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