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## G = C2×C8×D9order 288 = 25·32

### Direct product of C2×C8 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C8×D9
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — C2×C4×D9 — C2×C8×D9
 Lower central C9 — C2×C8×D9
 Upper central C1 — C2×C8

Generators and relations for C2×C8×D9
G = < a,b,c,d | a2=b8=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 384 in 114 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], S3 [×4], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C9, Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C2×C8, C2×C8 [×5], C22×C4, D9 [×4], C18, C18 [×2], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C22×C8, Dic9 [×2], C36 [×2], D18 [×6], C2×C18, S3×C8 [×4], C2×C3⋊C8, C2×C24, S3×C2×C4, C9⋊C8 [×2], C72 [×2], C4×D9 [×4], C2×Dic9, C2×C36, C22×D9, S3×C2×C8, C8×D9 [×4], C2×C9⋊C8, C2×C72, C2×C4×D9, C2×C8×D9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C8 [×4], C2×C4 [×6], C23, D6 [×3], C2×C8 [×6], C22×C4, D9, C4×S3 [×2], C22×S3, C22×C8, D18 [×3], S3×C8 [×2], S3×C2×C4, C4×D9 [×2], C22×D9, S3×C2×C8, C8×D9 [×2], C2×C4×D9, C2×C8×D9

Smallest permutation representation of C2×C8×D9
On 144 points
Generators in S144
(1 77)(2 78)(3 79)(4 80)(5 81)(6 73)(7 74)(8 75)(9 76)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 68 32 50 14 59 23 41)(2 69 33 51 15 60 24 42)(3 70 34 52 16 61 25 43)(4 71 35 53 17 62 26 44)(5 72 36 54 18 63 27 45)(6 64 28 46 10 55 19 37)(7 65 29 47 11 56 20 38)(8 66 30 48 12 57 21 39)(9 67 31 49 13 58 22 40)(73 136 100 118 82 127 91 109)(74 137 101 119 83 128 92 110)(75 138 102 120 84 129 93 111)(76 139 103 121 85 130 94 112)(77 140 104 122 86 131 95 113)(78 141 105 123 87 132 96 114)(79 142 106 124 88 133 97 115)(80 143 107 125 89 134 98 116)(81 144 108 126 90 135 99 117)
(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 76)(2 75)(3 74)(4 73)(5 81)(6 80)(7 79)(8 78)(9 77)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 90)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 99)(28 107)(29 106)(30 105)(31 104)(32 103)(33 102)(34 101)(35 100)(36 108)(37 116)(38 115)(39 114)(40 113)(41 112)(42 111)(43 110)(44 109)(45 117)(46 125)(47 124)(48 123)(49 122)(50 121)(51 120)(52 119)(53 118)(54 126)(55 134)(56 133)(57 132)(58 131)(59 130)(60 129)(61 128)(62 127)(63 135)(64 143)(65 142)(66 141)(67 140)(68 139)(69 138)(70 137)(71 136)(72 144)

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

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

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

96 conjugacy classes

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

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 S3 D6 D6 D9 C4×S3 C4×S3 D18 D18 S3×C8 C4×D9 C4×D9 C8×D9 kernel C2×C8×D9 C8×D9 C2×C9⋊C8 C2×C72 C2×C4×D9 C4×D9 C2×Dic9 C22×D9 D18 C2×C24 C24 C2×C12 C2×C8 C12 C2×C6 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 16 1 2 1 3 2 2 6 3 8 6 6 24

Matrix representation of C2×C8×D9 in GL3(𝔽73) generated by

 1 0 0 0 72 0 0 0 72
,
 63 0 0 0 1 0 0 0 1
,
 1 0 0 0 31 70 0 3 28
,
 72 0 0 0 70 45 0 42 3
G:=sub<GL(3,GF(73))| [1,0,0,0,72,0,0,0,72],[63,0,0,0,1,0,0,0,1],[1,0,0,0,31,3,0,70,28],[72,0,0,0,70,42,0,45,3] >;

C2×C8×D9 in GAP, Magma, Sage, TeX

C_2\times C_8\times D_9
% in TeX

G:=Group("C2xC8xD9");
// GroupNames label

G:=SmallGroup(288,110);
// by ID

G=gap.SmallGroup(288,110);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,58,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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