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 [×2], C3, C3, C4 [×2], C22, C6, C6 [×2], C6 [×3], C8 [×2], C2×C4, C9, C9, C32, C12 [×2], C12 [×2], C2×C6, C2×C6, C2×C8, C18, C18 [×2], C18 [×3], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], C2×C12, C2×C12, 3- 1+2, C36 [×2], C36 [×2], C2×C18, C2×C18, C3×C12 [×2], C62, C2×C3⋊C8, C2×C24, C2×3- 1+2, C2×3- 1+2 [×2], C9⋊C8 [×2], C2×C36, C2×C36, C3×C3⋊C8 [×2], C6×C12, C4×3- 1+2 [×2], C22×3- 1+2, C2×C9⋊C8, C6×C3⋊C8, C9⋊C24 [×2], C2×C4×3- 1+2, C2×C9⋊C24
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C2×C8, C3×S3, C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12, C3×Dic3 [×2], S3×C6, C2×C3⋊C8, C2×C24, C9⋊C6, C3×C3⋊C8 [×2], C6×Dic3, C9⋊C12 [×2], C2×C9⋊C6, C6×C3⋊C8, C9⋊C24 [×2], C2×C9⋊C12, C2×C9⋊C24
►On 144 points
Generators in S144
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 140)(74 141)(75 142)(76 143)(77 144)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)
(1 72 88 42 64 96 12 56 80)(2 89 65 13 81 49 43 73 57)(3 66 82 44 58 90 14 50 74)(4 83 59 15 75 67 45 91 51)(5 60 76 46 52 84 16 68 92)(6 77 53 9 93 61 47 85 69)(7 54 94 48 70 78 10 62 86)(8 95 71 11 87 55 41 79 63)(17 103 131 35 119 139 28 111 123)(18 132 120 29 124 104 36 140 112)(19 97 125 37 113 133 30 105 141)(20 126 114 31 142 98 38 134 106)(21 115 143 39 107 127 32 99 135)(22 144 108 25 136 116 40 128 100)(23 109 137 33 101 121 26 117 129)(24 138 102 27 130 110 34 122 118)
(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)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,140)(74,141)(75,142)(76,143)(77,144)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139), (1,72,88,42,64,96,12,56,80)(2,89,65,13,81,49,43,73,57)(3,66,82,44,58,90,14,50,74)(4,83,59,15,75,67,45,91,51)(5,60,76,46,52,84,16,68,92)(6,77,53,9,93,61,47,85,69)(7,54,94,48,70,78,10,62,86)(8,95,71,11,87,55,41,79,63)(17,103,131,35,119,139,28,111,123)(18,132,120,29,124,104,36,140,112)(19,97,125,37,113,133,30,105,141)(20,126,114,31,142,98,38,134,106)(21,115,143,39,107,127,32,99,135)(22,144,108,25,136,116,40,128,100)(23,109,137,33,101,121,26,117,129)(24,138,102,27,130,110,34,122,118), (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( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,140)(74,141)(75,142)(76,143)(77,144)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139), (1,72,88,42,64,96,12,56,80)(2,89,65,13,81,49,43,73,57)(3,66,82,44,58,90,14,50,74)(4,83,59,15,75,67,45,91,51)(5,60,76,46,52,84,16,68,92)(6,77,53,9,93,61,47,85,69)(7,54,94,48,70,78,10,62,86)(8,95,71,11,87,55,41,79,63)(17,103,131,35,119,139,28,111,123)(18,132,120,29,124,104,36,140,112)(19,97,125,37,113,133,30,105,141)(20,126,114,31,142,98,38,134,106)(21,115,143,39,107,127,32,99,135)(22,144,108,25,136,116,40,128,100)(23,109,137,33,101,121,26,117,129)(24,138,102,27,130,110,34,122,118), (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([(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112),(58,113),(59,114),(60,115),(61,116),(62,117),(63,118),(64,119),(65,120),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103),(73,140),(74,141),(75,142),(76,143),(77,144),(78,121),(79,122),(80,123),(81,124),(82,125),(83,126),(84,127),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,134),(92,135),(93,136),(94,137),(95,138),(96,139)], [(1,72,88,42,64,96,12,56,80),(2,89,65,13,81,49,43,73,57),(3,66,82,44,58,90,14,50,74),(4,83,59,15,75,67,45,91,51),(5,60,76,46,52,84,16,68,92),(6,77,53,9,93,61,47,85,69),(7,54,94,48,70,78,10,62,86),(8,95,71,11,87,55,41,79,63),(17,103,131,35,119,139,28,111,123),(18,132,120,29,124,104,36,140,112),(19,97,125,37,113,133,30,105,141),(20,126,114,31,142,98,38,134,106),(21,115,143,39,107,127,32,99,135),(22,144,108,25,136,116,40,128,100),(23,109,137,33,101,121,26,117,129),(24,138,102,27,130,110,34,122,118)], [(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