direct product, metacyclic, supersoluble, monomial
Aliases: C2×C9⋊C24, C18⋊C24, C36.3C12, C62.3Dic3, C9⋊C8⋊7C6, C9⋊2(C2×C24), (C2×C36).4C6, C4.3(C9⋊C12), C12.94(S3×C6), (C2×C18).1C12, C18.5(C2×C12), (C6×C12).18S3, C36.15(C2×C6), (C3×C12).62D6, C6.14(C6×Dic3), (C3×C12).9Dic3, (C2×3- 1+2)⋊C8, C22.2(C9⋊C12), C12.15(C3×Dic3), 3- 1+2⋊2(C2×C8), (C4×3- 1+2).3C4, (C22×3- 1+2).1C4, (C4×3- 1+2).14C22, (C2×C9⋊C8)⋊C3, C3.3(C6×C3⋊C8), C6.6(C3×C3⋊C8), C32.(C2×C3⋊C8), C4.14(C2×C9⋊C6), C2.1(C2×C9⋊C12), (C3×C6).5(C3⋊C8), (C2×C4).5(C9⋊C6), (C2×C12).33(C3×S3), (C2×C6).16(C3×Dic3), (C3×C6).10(C2×Dic3), (C2×C4×3- 1+2).4C2, (C2×3- 1+2).5(C2×C4), SmallGroup(432,142)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C2×C9⋊C24 |
Generators and relations for C2×C9⋊C24
G = < a,b,c | a2=b9=c24=1, ab=ba, ac=ca, cbc-1=b2 >
Subgroups: 158 in 74 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C9, C9, C32, C12, C12, C2×C6, C2×C6, C2×C8, C18, C18, C18, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, 3- 1+2, C36, C36, C2×C18, C2×C18, C3×C12, C62, C2×C3⋊C8, C2×C24, C2×3- 1+2, C2×3- 1+2, C9⋊C8, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C4×3- 1+2, C22×3- 1+2, C2×C9⋊C8, C6×C3⋊C8, C9⋊C24, C2×C4×3- 1+2, C2×C9⋊C24
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊C8, C24, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C2×C3⋊C8, C2×C24, C9⋊C6, C3×C3⋊C8, C6×Dic3, C9⋊C12, C2×C9⋊C6, C6×C3⋊C8, C9⋊C24, C2×C9⋊C12, C2×C9⋊C24
►On 144 points
Generators in S144
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 141)(80 142)(81 143)(82 144)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)
(1 53 94 19 69 78 26 61 86)(2 95 70 27 87 54 20 79 62)(3 71 88 21 63 96 28 55 80)(4 89 64 29 81 72 22 73 56)(5 65 82 23 57 90 30 49 74)(6 83 58 31 75 66 24 91 50)(7 59 76 17 51 84 32 67 92)(8 77 52 25 93 60 18 85 68)(9 139 114 48 131 98 38 123 106)(10 115 132 39 107 140 41 99 124)(11 133 108 42 125 116 40 141 100)(12 109 126 33 101 134 43 117 142)(13 127 102 44 143 110 34 135 118)(14 103 144 35 119 128 45 111 136)(15 121 120 46 137 104 36 129 112)(16 97 138 37 113 122 47 105 130)
(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)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134), (1,53,94,19,69,78,26,61,86)(2,95,70,27,87,54,20,79,62)(3,71,88,21,63,96,28,55,80)(4,89,64,29,81,72,22,73,56)(5,65,82,23,57,90,30,49,74)(6,83,58,31,75,66,24,91,50)(7,59,76,17,51,84,32,67,92)(8,77,52,25,93,60,18,85,68)(9,139,114,48,131,98,38,123,106)(10,115,132,39,107,140,41,99,124)(11,133,108,42,125,116,40,141,100)(12,109,126,33,101,134,43,117,142)(13,127,102,44,143,110,34,135,118)(14,103,144,35,119,128,45,111,136)(15,121,120,46,137,104,36,129,112)(16,97,138,37,113,122,47,105,130), (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( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,135)(74,136)(75,137)(76,138)(77,139)(78,140)(79,141)(80,142)(81,143)(82,144)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134), (1,53,94,19,69,78,26,61,86)(2,95,70,27,87,54,20,79,62)(3,71,88,21,63,96,28,55,80)(4,89,64,29,81,72,22,73,56)(5,65,82,23,57,90,30,49,74)(6,83,58,31,75,66,24,91,50)(7,59,76,17,51,84,32,67,92)(8,77,52,25,93,60,18,85,68)(9,139,114,48,131,98,38,123,106)(10,115,132,39,107,140,41,99,124)(11,133,108,42,125,116,40,141,100)(12,109,126,33,101,134,43,117,142)(13,127,102,44,143,110,34,135,118)(14,103,144,35,119,128,45,111,136)(15,121,120,46,137,104,36,129,112)(16,97,138,37,113,122,47,105,130), (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([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,48),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,97),(60,98),(61,99),(62,100),(63,101),(64,102),(65,103),(66,104),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110),(73,135),(74,136),(75,137),(76,138),(77,139),(78,140),(79,141),(80,142),(81,143),(82,144),(83,121),(84,122),(85,123),(86,124),(87,125),(88,126),(89,127),(90,128),(91,129),(92,130),(93,131),(94,132),(95,133),(96,134)], [(1,53,94,19,69,78,26,61,86),(2,95,70,27,87,54,20,79,62),(3,71,88,21,63,96,28,55,80),(4,89,64,29,81,72,22,73,56),(5,65,82,23,57,90,30,49,74),(6,83,58,31,75,66,24,91,50),(7,59,76,17,51,84,32,67,92),(8,77,52,25,93,60,18,85,68),(9,139,114,48,131,98,38,123,106),(10,115,132,39,107,140,41,99,124),(11,133,108,42,125,116,40,141,100),(12,109,126,33,101,134,43,117,142),(13,127,102,44,143,110,34,135,118),(14,103,144,35,119,128,45,111,136),(15,121,120,46,137,104,36,129,112),(16,97,138,37,113,122,47,105,130)], [(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 | 6A | 6B | 6C | 6D | ··· | 6I | 8A | ··· | 8H | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 18A | ··· | 18I | 24A | ··· | 24P | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 9 | ··· | 9 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 6 | ··· | 6 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | - | + | - | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | Dic3 | D6 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | S3×C6 | C3×Dic3 | C3×C3⋊C8 | C9⋊C6 | C9⋊C12 | C2×C9⋊C6 | C9⋊C12 | C9⋊C24 |
| kernel | C2×C9⋊C24 | C9⋊C24 | C2×C4×3- 1+2 | C2×C9⋊C8 | C4×3- 1+2 | C22×3- 1+2 | C9⋊C8 | C2×C36 | C2×3- 1+2 | C36 | C2×C18 | C18 | C6×C12 | C3×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C6 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 4 |
Matrix representation of C2×C9⋊C24 ►in GL10(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 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 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 72 | 71 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 72 | 72 | 72 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 72 | 0 | 0 |
| 22 | 38 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 51 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 63 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 51 | 61 |
| 0 | 0 | 0 | 0 | 63 | 51 | 0 | 0 | 10 | 22 |
| 0 | 0 | 0 | 0 | 63 | 51 | 10 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 61 | 12 | 63 | 0 | 0 |
G:=sub<GL(10,GF(73))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,71,72,72,72,72,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0],[22,16,0,0,0,0,0,0,0,0,38,51,0,0,0,0,0,0,0,0,0,0,64,11,0,0,0,0,0,0,0,0,20,9,0,0,0,0,0,0,0,0,0,0,12,63,12,63,63,0,0,0,0,0,51,61,0,51,51,61,0,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,22,63,0,0,0,0,0,0,51,10,0,0,0,0,0,0,0,0,61,22,0,0] >;
C2×C9⋊C24 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes C_{24} % in TeX
G:=Group("C2xC9:C24"); // GroupNames label
G:=SmallGroup(432,142);
// by ID
G=gap.SmallGroup(432,142);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,80,10085,2035,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^2=b^9=c^24=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations