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## G = C9×C24⋊C2order 432 = 24·33

### Direct product of C9 and C24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C9×C24⋊C2
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — C9×D12 — C9×C24⋊C2
 Lower central C3 — C6 — C12 — C9×C24⋊C2
 Upper central C1 — C18 — C36 — C72

Generators and relations for C9×C24⋊C2
G = < a,b,c | a9=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Subgroups: 172 in 68 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, D4, Q8, C9, C9, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, SD16, C18, C18 [×2], C3×S3, C3×C6, C24 [×2], C24, Dic6, D12, C3×D4, C3×Q8, C3×C9, C36, C36 [×2], C2×C18, C3×Dic3, C3×C12, S3×C6, C24⋊C2, C3×SD16, S3×C9, C3×C18, C72, C72, D4×C9, Q8×C9, C3×C24, C3×Dic6, C3×D12, C9×Dic3, C3×C36, S3×C18, C9×SD16, C3×C24⋊C2, C3×C72, C9×Dic6, C9×D12, C9×C24⋊C2
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, C9, D6, C2×C6, SD16, C18 [×3], C3×S3, D12, C3×D4, C2×C18, S3×C6, C24⋊C2, C3×SD16, S3×C9, D4×C9, C3×D12, S3×C18, C9×SD16, C3×C24⋊C2, C9×D12, C9×C24⋊C2

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

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

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

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

135 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 9A ··· 9F 9G ··· 9L 12A ··· 12H 12I 12J 18A ··· 18F 18G ··· 18L 18M ··· 18R 24A ··· 24P 36A ··· 36R 36S ··· 36X 72A ··· 72AJ order 1 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 8 8 9 ··· 9 9 ··· 9 12 ··· 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 24 ··· 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 12 1 1 2 2 2 2 12 1 1 2 2 2 12 12 2 2 1 ··· 1 2 ··· 2 2 ··· 2 12 12 1 ··· 1 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2 12 ··· 12 2 ··· 2

135 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 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 S3 D4 D6 SD16 C3×S3 D12 C3×D4 S3×C6 C24⋊C2 C3×SD16 S3×C9 D4×C9 C3×D12 S3×C18 C9×SD16 C3×C24⋊C2 C9×D12 C9×C24⋊C2 kernel C9×C24⋊C2 C3×C72 C9×Dic6 C9×D12 C3×C24⋊C2 C3×C24 C3×Dic6 C3×D12 C24⋊C2 C24 Dic6 D12 C72 C3×C18 C36 C3×C9 C24 C18 C3×C6 C12 C9 C32 C8 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 6 6 6 6 1 1 1 2 2 2 2 2 4 4 6 6 4 6 12 8 12 24

Matrix representation of C9×C24⋊C2 in GL4(𝔽73) generated by

 32 0 0 0 0 32 0 0 0 0 64 0 0 0 0 64
,
 3 0 0 0 0 49 0 0 0 0 6 67 0 0 6 6
,
 0 1 0 0 1 0 0 0 0 0 16 16 0 0 16 57
G:=sub<GL(4,GF(73))| [32,0,0,0,0,32,0,0,0,0,64,0,0,0,0,64],[3,0,0,0,0,49,0,0,0,0,6,6,0,0,67,6],[0,1,0,0,1,0,0,0,0,0,16,16,0,0,16,57] >;

C9×C24⋊C2 in GAP, Magma, Sage, TeX

C_9\times C_{24}\rtimes C_2
% in TeX

G:=Group("C9xC24:C2");
// GroupNames label

G:=SmallGroup(432,111);
// by ID

G=gap.SmallGroup(432,111);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,197,92,142,2355,192,14118]);
// Polycyclic

G:=Group<a,b,c|a^9=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^11>;
// generators/relations

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