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G = C2×C9⋊C24order 432 = 24·33

Direct product of C2 and C9⋊C24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C9⋊C24
 Chief series C1 — C3 — C9 — C18 — C36 — C4×3- 1+2 — C9⋊C24 — C2×C9⋊C24
 Lower central C9 — C2×C9⋊C24
 Upper central C1 — C2×C4

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

Smallest permutation representation of 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

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