direct product, metacyclic, supersoluble, monomial
Aliases: C4×C9⋊C12, C36⋊2C12, Dic9⋊2C12, C62.22D6, 3- 1+2⋊C42, C9⋊(C4×C12), (C4×Dic9)⋊C3, (C2×C36).5C6, C6.17(S3×C12), C18.7(C2×C12), (C6×C12).19S3, C32.(C4×Dic3), (C2×Dic9).4C6, C3.3(Dic3×C12), C6.16(C6×Dic3), C12.16(C3×Dic3), (C3×C12).10Dic3, (C4×3- 1+2)⋊2C4, (C22×3- 1+2).2C22, C2.2(C4×C9⋊C6), C2.2(C2×C9⋊C12), (C2×C9⋊C12).4C2, (C2×C4).6(C9⋊C6), (C3×C6).17(C4×S3), (C2×C18).2(C2×C6), (C2×C6).42(S3×C6), C22.3(C2×C9⋊C6), (C2×C12).34(C3×S3), (C3×C6).12(C2×Dic3), (C2×C4×3- 1+2).5C2, (C2×3- 1+2).7(C2×C4), SmallGroup(432,144)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C4×C9⋊C12 |
Generators and relations for C4×C9⋊C12
G = < a,b,c | a4=b9=c12=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 262 in 98 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, 3- 1+2, Dic9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C4×Dic3, C4×C12, C2×3- 1+2, C2×3- 1+2, C2×Dic9, C2×C36, C2×C36, C6×Dic3, C6×C12, C9⋊C12, C4×3- 1+2, C22×3- 1+2, C4×Dic9, Dic3×C12, C2×C9⋊C12, C2×C4×3- 1+2, C4×C9⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, C3×S3, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C4×Dic3, C4×C12, C9⋊C6, S3×C12, C6×Dic3, C9⋊C12, C2×C9⋊C6, Dic3×C12, C4×C9⋊C6, C2×C9⋊C12, C4×C9⋊C12
►On 144 points
Generators in S144
(1 11 26 46)(2 12 27 47)(3 9 28 48)(4 10 25 45)(5 13 43 31)(6 14 44 32)(7 15 41 29)(8 16 42 30)(17 38 34 23)(18 39 35 24)(19 40 36 21)(20 37 33 22)(49 131 65 91)(50 132 66 92)(51 121 67 93)(52 122 68 94)(53 123 69 95)(54 124 70 96)(55 125 71 85)(56 126 72 86)(57 127 61 87)(58 128 62 88)(59 129 63 89)(60 130 64 90)(73 120 141 107)(74 109 142 108)(75 110 143 97)(76 111 144 98)(77 112 133 99)(78 113 134 100)(79 114 135 101)(80 115 136 102)(81 116 137 103)(82 117 138 104)(83 118 139 105)(84 119 140 106)
(1 142 51 15 138 55 20 134 59)(2 56 143 17 52 135 16 60 139)(3 136 57 13 144 49 18 140 53)(4 50 137 19 58 141 14 54 133)(5 111 91 24 119 95 48 115 87)(6 96 112 45 92 116 21 88 120)(7 117 85 22 113 89 46 109 93)(8 90 118 47 86 110 23 94 114)(9 102 127 43 98 131 39 106 123)(10 132 103 40 128 107 44 124 99)(11 108 121 41 104 125 37 100 129)(12 126 97 38 122 101 42 130 105)(25 66 81 36 62 73 32 70 77)(26 74 67 29 82 71 33 78 63)(27 72 75 34 68 79 30 64 83)(28 80 61 31 76 65 35 84 69)
(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)
G:=sub<Sym(144)| (1,11,26,46)(2,12,27,47)(3,9,28,48)(4,10,25,45)(5,13,43,31)(6,14,44,32)(7,15,41,29)(8,16,42,30)(17,38,34,23)(18,39,35,24)(19,40,36,21)(20,37,33,22)(49,131,65,91)(50,132,66,92)(51,121,67,93)(52,122,68,94)(53,123,69,95)(54,124,70,96)(55,125,71,85)(56,126,72,86)(57,127,61,87)(58,128,62,88)(59,129,63,89)(60,130,64,90)(73,120,141,107)(74,109,142,108)(75,110,143,97)(76,111,144,98)(77,112,133,99)(78,113,134,100)(79,114,135,101)(80,115,136,102)(81,116,137,103)(82,117,138,104)(83,118,139,105)(84,119,140,106), (1,142,51,15,138,55,20,134,59)(2,56,143,17,52,135,16,60,139)(3,136,57,13,144,49,18,140,53)(4,50,137,19,58,141,14,54,133)(5,111,91,24,119,95,48,115,87)(6,96,112,45,92,116,21,88,120)(7,117,85,22,113,89,46,109,93)(8,90,118,47,86,110,23,94,114)(9,102,127,43,98,131,39,106,123)(10,132,103,40,128,107,44,124,99)(11,108,121,41,104,125,37,100,129)(12,126,97,38,122,101,42,130,105)(25,66,81,36,62,73,32,70,77)(26,74,67,29,82,71,33,78,63)(27,72,75,34,68,79,30,64,83)(28,80,61,31,76,65,35,84,69), (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)>;
G:=Group( (1,11,26,46)(2,12,27,47)(3,9,28,48)(4,10,25,45)(5,13,43,31)(6,14,44,32)(7,15,41,29)(8,16,42,30)(17,38,34,23)(18,39,35,24)(19,40,36,21)(20,37,33,22)(49,131,65,91)(50,132,66,92)(51,121,67,93)(52,122,68,94)(53,123,69,95)(54,124,70,96)(55,125,71,85)(56,126,72,86)(57,127,61,87)(58,128,62,88)(59,129,63,89)(60,130,64,90)(73,120,141,107)(74,109,142,108)(75,110,143,97)(76,111,144,98)(77,112,133,99)(78,113,134,100)(79,114,135,101)(80,115,136,102)(81,116,137,103)(82,117,138,104)(83,118,139,105)(84,119,140,106), (1,142,51,15,138,55,20,134,59)(2,56,143,17,52,135,16,60,139)(3,136,57,13,144,49,18,140,53)(4,50,137,19,58,141,14,54,133)(5,111,91,24,119,95,48,115,87)(6,96,112,45,92,116,21,88,120)(7,117,85,22,113,89,46,109,93)(8,90,118,47,86,110,23,94,114)(9,102,127,43,98,131,39,106,123)(10,132,103,40,128,107,44,124,99)(11,108,121,41,104,125,37,100,129)(12,126,97,38,122,101,42,130,105)(25,66,81,36,62,73,32,70,77)(26,74,67,29,82,71,33,78,63)(27,72,75,34,68,79,30,64,83)(28,80,61,31,76,65,35,84,69), (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) );
G=PermutationGroup([[(1,11,26,46),(2,12,27,47),(3,9,28,48),(4,10,25,45),(5,13,43,31),(6,14,44,32),(7,15,41,29),(8,16,42,30),(17,38,34,23),(18,39,35,24),(19,40,36,21),(20,37,33,22),(49,131,65,91),(50,132,66,92),(51,121,67,93),(52,122,68,94),(53,123,69,95),(54,124,70,96),(55,125,71,85),(56,126,72,86),(57,127,61,87),(58,128,62,88),(59,129,63,89),(60,130,64,90),(73,120,141,107),(74,109,142,108),(75,110,143,97),(76,111,144,98),(77,112,133,99),(78,113,134,100),(79,114,135,101),(80,115,136,102),(81,116,137,103),(82,117,138,104),(83,118,139,105),(84,119,140,106)], [(1,142,51,15,138,55,20,134,59),(2,56,143,17,52,135,16,60,139),(3,136,57,13,144,49,18,140,53),(4,50,137,19,58,141,14,54,133),(5,111,91,24,119,95,48,115,87),(6,96,112,45,92,116,21,88,120),(7,117,85,22,113,89,46,109,93),(8,90,118,47,86,110,23,94,114),(9,102,127,43,98,131,39,106,123),(10,132,103,40,128,107,44,124,99),(11,108,121,41,104,125,37,100,129),(12,126,97,38,122,101,42,130,105),(25,66,81,36,62,73,32,70,77),(26,74,67,29,82,71,33,78,63),(27,72,75,34,68,79,30,64,83),(28,80,61,31,76,65,35,84,69)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 6D | ··· | 6I | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | ··· | 12AB | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 9 | ··· | 9 | 6 | ··· | 6 | 6 | ··· | 6 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | + | - | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | C3×S3 | C4×S3 | C3×Dic3 | S3×C6 | S3×C12 | C9⋊C6 | C9⋊C12 | C2×C9⋊C6 | C4×C9⋊C6 |
| kernel | C4×C9⋊C12 | C2×C9⋊C12 | C2×C4×3- 1+2 | C4×Dic9 | C9⋊C12 | C4×3- 1+2 | C2×Dic9 | C2×C36 | Dic9 | C36 | C6×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 1 | 4 |
Matrix representation of C4×C9⋊C12 ►in GL10(𝔽37)
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 36 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 36 | 36 | 36 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 36 | 0 | 0 |
| 1 | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 29 | 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 4 | 0 | 0 | 25 | 33 |
| 0 | 0 | 0 | 0 | 0 | 29 | 0 | 0 | 8 | 12 |
| 0 | 0 | 0 | 0 | 8 | 0 | 4 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 4 | 25 | 33 | 0 | 0 |
G:=sub<GL(10,GF(37))| [6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,36,1,0,0,0,0,0,0,0,0,35,36,36,36,36,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0],[1,29,0,0,0,0,0,0,0,0,28,36,0,0,0,0,0,0,0,0,0,0,21,31,0,0,0,0,0,0,0,0,10,16,0,0,0,0,0,0,0,0,0,0,8,12,12,0,8,12,0,0,0,0,4,29,4,29,0,4,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,0,0,29,33,0,0,0,0,0,0,25,8,0,0,0,0,0,0,0,0,33,12,0,0] >;
C4×C9⋊C12 in GAP, Magma, Sage, TeX
C_4\times C_9\rtimes C_{12} % in TeX
G:=Group("C4xC9:C12"); // GroupNames label
G:=SmallGroup(432,144);
// by ID
G=gap.SmallGroup(432,144);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,10085,2035,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^4=b^9=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations