direct product, metabelian, nilpotent (class 2), monomial
Aliases: C22×C4×3- 1+2, C62.13C12, C12.44C62, C36⋊6(C2×C6), (C2×C36)⋊8C6, C18⋊3(C2×C12), (C2×C18)⋊8C12, C6.18(C6×C12), (C6×C12).20C6, C9⋊3(C22×C12), (C22×C36)⋊3C3, (C2×C62).15C6, C6.12(C2×C62), (C2×C6).33C62, C62.38(C2×C6), C32.(C22×C12), (C22×C18).10C6, C18.11(C22×C6), (C22×C12).7C32, C23.4(C2×3- 1+2), C2.1(C23×3- 1+2), (C23×3- 1+2).4C2, (C2×3- 1+2).11C23, C22.5(C22×3- 1+2), (C22×3- 1+2).16C22, C3.2(C2×C6×C12), (C2×C6×C12).3C3, (C2×C18).18(C2×C6), (C2×C6).18(C3×C12), (C3×C12).70(C2×C6), (C2×C12).33(C3×C6), (C3×C6).36(C2×C12), (C22×C6).23(C3×C6), (C3×C6).30(C22×C6), SmallGroup(432,402)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C9, C32, C12, C12, C2×C6, C2×C6, C22×C4, C18, C3×C6, C3×C6, C2×C12, C2×C12, C22×C6, C22×C6, 3- 1+2, C36, C2×C18, C3×C12, C62, C22×C12, C22×C12, C2×3- 1+2, C2×3- 1+2, C2×C36, C22×C18, C6×C12, C2×C62, C4×3- 1+2, C22×3- 1+2, C22×C36, C2×C6×C12, C2×C4×3- 1+2, C23×3- 1+2, C22×C4×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, C22×C4, C3×C6, C2×C12, C22×C6, 3- 1+2, C3×C12, C62, C22×C12, C2×3- 1+2, C6×C12, C2×C62, C4×3- 1+2, C22×3- 1+2, C2×C6×C12, C2×C4×3- 1+2, C23×3- 1+2, 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