direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C42×3- 1+2, C36⋊4C12, C3.2C122, C122.3C3, (C4×C36)⋊3C3, C9⋊2(C4×C12), C32.(C4×C12), C6.11(C6×C12), (C6×C12).17C6, (C2×C36).10C6, C12.14(C3×C12), (C3×C12).12C12, C18.10(C2×C12), (C2×C6).25C62, (C4×C12).7C32, C62.32(C2×C6), C22.2(C22×3- 1+2), (C22×3- 1+2).13C22, (C2×C18).15(C2×C6), (C3×C6).27(C2×C12), (C2×C12).29(C3×C6), C2.1(C2×C4×3- 1+2), (C2×C4×3- 1+2).10C2, (C2×C4).4(C2×3- 1+2), (C2×3- 1+2).10(C2×C4), SmallGroup(432,202)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C6 — C2×C6 — C62 — C22×3- 1+2 — C2×C4×3- 1+2 — C42×3- 1+2 |
Generators and relations for C42×3- 1+2
G = < a,b,c,d | a4=b4=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Subgroups: 150 in 120 conjugacy classes, 105 normal (12 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, C42, C18, C3×C6, C2×C12, C2×C12, 3- 1+2, C36, C2×C18, C3×C12, C62, C4×C12, C4×C12, C2×3- 1+2, C2×C36, C6×C12, C4×3- 1+2, C22×3- 1+2, C4×C36, C122, C2×C4×3- 1+2, C42×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C42, C3×C6, C2×C12, 3- 1+2, C3×C12, C62, C4×C12, C2×3- 1+2, C6×C12, C4×3- 1+2, C22×3- 1+2, C122, C2×C4×3- 1+2, C42×3- 1+2
►On 144 points
Generators in S144
(1 128 52 142)(2 129 53 143)(3 130 54 144)(4 131 46 136)(5 132 47 137)(6 133 48 138)(7 134 49 139)(8 135 50 140)(9 127 51 141)(10 39 20 34)(11 40 21 35)(12 41 22 36)(13 42 23 28)(14 43 24 29)(15 44 25 30)(16 45 26 31)(17 37 27 32)(18 38 19 33)(55 105 69 91)(56 106 70 92)(57 107 71 93)(58 108 72 94)(59 100 64 95)(60 101 65 96)(61 102 66 97)(62 103 67 98)(63 104 68 99)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)
(1 56 40 88)(2 57 41 89)(3 58 42 90)(4 59 43 82)(5 60 44 83)(6 61 45 84)(7 62 37 85)(8 63 38 86)(9 55 39 87)(10 123 141 91)(11 124 142 92)(12 125 143 93)(13 126 144 94)(14 118 136 95)(15 119 137 96)(16 120 138 97)(17 121 139 98)(18 122 140 99)(19 113 135 104)(20 114 127 105)(21 115 128 106)(22 116 129 107)(23 117 130 108)(24 109 131 100)(25 110 132 101)(26 111 133 102)(27 112 134 103)(28 81 54 72)(29 73 46 64)(30 74 47 65)(31 75 48 66)(32 76 49 67)(33 77 50 68)(34 78 51 69)(35 79 52 70)(36 80 53 71)
(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 13 16)(12 18 15)(19 25 22)(20 23 26)(28 31 34)(30 36 33)(38 44 41)(39 42 45)(47 53 50)(48 51 54)(55 58 61)(57 63 60)(65 71 68)(66 69 72)(74 80 77)(75 78 81)(83 89 86)(84 87 90)(91 94 97)(93 99 96)(101 107 104)(102 105 108)(110 116 113)(111 114 117)(119 125 122)(120 123 126)(127 130 133)(129 135 132)(137 143 140)(138 141 144)
G:=sub<Sym(144)| (1,128,52,142)(2,129,53,143)(3,130,54,144)(4,131,46,136)(5,132,47,137)(6,133,48,138)(7,134,49,139)(8,135,50,140)(9,127,51,141)(10,39,20,34)(11,40,21,35)(12,41,22,36)(13,42,23,28)(14,43,24,29)(15,44,25,30)(16,45,26,31)(17,37,27,32)(18,38,19,33)(55,105,69,91)(56,106,70,92)(57,107,71,93)(58,108,72,94)(59,100,64,95)(60,101,65,96)(61,102,66,97)(62,103,67,98)(63,104,68,99)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117), (1,56,40,88)(2,57,41,89)(3,58,42,90)(4,59,43,82)(5,60,44,83)(6,61,45,84)(7,62,37,85)(8,63,38,86)(9,55,39,87)(10,123,141,91)(11,124,142,92)(12,125,143,93)(13,126,144,94)(14,118,136,95)(15,119,137,96)(16,120,138,97)(17,121,139,98)(18,122,140,99)(19,113,135,104)(20,114,127,105)(21,115,128,106)(22,116,129,107)(23,117,130,108)(24,109,131,100)(25,110,132,101)(26,111,133,102)(27,112,134,103)(28,81,54,72)(29,73,46,64)(30,74,47,65)(31,75,48,66)(32,76,49,67)(33,77,50,68)(34,78,51,69)(35,79,52,70)(36,80,53,71), (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)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(28,31,34)(30,36,33)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(55,58,61)(57,63,60)(65,71,68)(66,69,72)(74,80,77)(75,78,81)(83,89,86)(84,87,90)(91,94,97)(93,99,96)(101,107,104)(102,105,108)(110,116,113)(111,114,117)(119,125,122)(120,123,126)(127,130,133)(129,135,132)(137,143,140)(138,141,144)>;
G:=Group( (1,128,52,142)(2,129,53,143)(3,130,54,144)(4,131,46,136)(5,132,47,137)(6,133,48,138)(7,134,49,139)(8,135,50,140)(9,127,51,141)(10,39,20,34)(11,40,21,35)(12,41,22,36)(13,42,23,28)(14,43,24,29)(15,44,25,30)(16,45,26,31)(17,37,27,32)(18,38,19,33)(55,105,69,91)(56,106,70,92)(57,107,71,93)(58,108,72,94)(59,100,64,95)(60,101,65,96)(61,102,66,97)(62,103,67,98)(63,104,68,99)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117), (1,56,40,88)(2,57,41,89)(3,58,42,90)(4,59,43,82)(5,60,44,83)(6,61,45,84)(7,62,37,85)(8,63,38,86)(9,55,39,87)(10,123,141,91)(11,124,142,92)(12,125,143,93)(13,126,144,94)(14,118,136,95)(15,119,137,96)(16,120,138,97)(17,121,139,98)(18,122,140,99)(19,113,135,104)(20,114,127,105)(21,115,128,106)(22,116,129,107)(23,117,130,108)(24,109,131,100)(25,110,132,101)(26,111,133,102)(27,112,134,103)(28,81,54,72)(29,73,46,64)(30,74,47,65)(31,75,48,66)(32,76,49,67)(33,77,50,68)(34,78,51,69)(35,79,52,70)(36,80,53,71), (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)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(28,31,34)(30,36,33)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(55,58,61)(57,63,60)(65,71,68)(66,69,72)(74,80,77)(75,78,81)(83,89,86)(84,87,90)(91,94,97)(93,99,96)(101,107,104)(102,105,108)(110,116,113)(111,114,117)(119,125,122)(120,123,126)(127,130,133)(129,135,132)(137,143,140)(138,141,144) );
G=PermutationGroup([[(1,128,52,142),(2,129,53,143),(3,130,54,144),(4,131,46,136),(5,132,47,137),(6,133,48,138),(7,134,49,139),(8,135,50,140),(9,127,51,141),(10,39,20,34),(11,40,21,35),(12,41,22,36),(13,42,23,28),(14,43,24,29),(15,44,25,30),(16,45,26,31),(17,37,27,32),(18,38,19,33),(55,105,69,91),(56,106,70,92),(57,107,71,93),(58,108,72,94),(59,100,64,95),(60,101,65,96),(61,102,66,97),(62,103,67,98),(63,104,68,99),(73,118,82,109),(74,119,83,110),(75,120,84,111),(76,121,85,112),(77,122,86,113),(78,123,87,114),(79,124,88,115),(80,125,89,116),(81,126,90,117)], [(1,56,40,88),(2,57,41,89),(3,58,42,90),(4,59,43,82),(5,60,44,83),(6,61,45,84),(7,62,37,85),(8,63,38,86),(9,55,39,87),(10,123,141,91),(11,124,142,92),(12,125,143,93),(13,126,144,94),(14,118,136,95),(15,119,137,96),(16,120,138,97),(17,121,139,98),(18,122,140,99),(19,113,135,104),(20,114,127,105),(21,115,128,106),(22,116,129,107),(23,117,130,108),(24,109,131,100),(25,110,132,101),(26,111,133,102),(27,112,134,103),(28,81,54,72),(29,73,46,64),(30,74,47,65),(31,75,48,66),(32,76,49,67),(33,77,50,68),(34,78,51,69),(35,79,52,70),(36,80,53,71)], [(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),(10,13,16),(12,18,15),(19,25,22),(20,23,26),(28,31,34),(30,36,33),(38,44,41),(39,42,45),(47,53,50),(48,51,54),(55,58,61),(57,63,60),(65,71,68),(66,69,72),(74,80,77),(75,78,81),(83,89,86),(84,87,90),(91,94,97),(93,99,96),(101,107,104),(102,105,108),(110,116,113),(111,114,117),(119,125,122),(120,123,126),(127,130,133),(129,135,132),(137,143,140),(138,141,144)]])
176 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6L | 9A | ··· | 9F | 12A | ··· | 12X | 12Y | ··· | 12AV | 18A | ··· | 18R | 36A | ··· | 36BT |
| order | 1 | 2 | 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 | 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 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | 3- 1+2 | C2×3- 1+2 | C4×3- 1+2 |
| kernel | C42×3- 1+2 | C2×C4×3- 1+2 | C4×C36 | C122 | C4×3- 1+2 | C2×C36 | C6×C12 | C36 | C3×C12 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 6 | 2 | 12 | 18 | 6 | 72 | 24 | 2 | 6 | 24 |
Matrix representation of C42×3- 1+2 ►in GL5(𝔽37)
| 31 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 6 | 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 | 26 | 9 | 0 |
| 0 | 0 | 10 | 11 | 10 |
| 0 | 0 | 0 | 27 | 0 |
| 26 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 26 | 10 | 0 |
| 0 | 0 | 36 | 0 | 26 |
G:=sub<GL(5,GF(37))| [31,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,26,10,0,0,0,9,11,27,0,0,0,10,0],[26,0,0,0,0,0,26,0,0,0,0,0,1,26,36,0,0,0,10,0,0,0,0,0,26] >;
C42×3- 1+2 in GAP, Magma, Sage, TeX
C_4^2\times 3_-^{1+2} % in TeX
G:=Group("C4^2xES-(3,1)"); // GroupNames label
G:=SmallGroup(432,202);
// by ID
G=gap.SmallGroup(432,202);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,512,772,1109]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations