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## G = C3×C8⋊D9order 432 = 24·33

### Direct product of C3 and C8⋊D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×C8⋊D9
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C12×D9 — C3×C8⋊D9
 Lower central C9 — C18 — C3×C8⋊D9
 Upper central C1 — C12 — C24

Generators and relations for C3×C8⋊D9
G = < a,b,c,d | a3=b8=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 222 in 68 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C3×M4(2), C3×D9, C3×C18, C9⋊C8, C72, C72, C4×D9, C3×C3⋊C8, C3×C24, S3×C12, C3×Dic9, C3×C36, C6×D9, C8⋊D9, C3×C8⋊S3, C3×C9⋊C8, C3×C72, C12×D9, C3×C8⋊D9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, M4(2), D9, C3×S3, C4×S3, C2×C12, D18, S3×C6, C8⋊S3, C3×M4(2), C3×D9, C4×D9, S3×C12, C6×D9, C8⋊D9, C3×C8⋊S3, C12×D9, C3×C8⋊D9

Smallest permutation representation of C3×C8⋊D9
On 144 points
Generators in S144
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)
(1 64 28 46 10 55 19 37)(2 65 29 47 11 56 20 38)(3 66 30 48 12 57 21 39)(4 67 31 49 13 58 22 40)(5 68 32 50 14 59 23 41)(6 69 33 51 15 60 24 42)(7 70 34 52 16 61 25 43)(8 71 35 53 17 62 26 44)(9 72 36 54 18 63 27 45)(73 127 100 109 82 136 91 118)(74 128 101 110 83 137 92 119)(75 129 102 111 84 138 93 120)(76 130 103 112 85 139 94 121)(77 131 104 113 86 140 95 122)(78 132 105 114 87 141 96 123)(79 133 106 115 88 142 97 124)(80 134 107 116 89 143 98 125)(81 135 108 117 90 144 99 126)
(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 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 81)(9 80)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 90)(18 89)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 99)(27 98)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 108)(36 107)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 117)(45 116)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 126)(54 125)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 135)(63 134)(64 142)(65 141)(66 140)(67 139)(68 138)(69 137)(70 136)(71 144)(72 143)

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

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

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

126 conjugacy classes

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

126 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 C4 C4 C6 C6 C6 C12 C12 S3 D6 M4(2) D9 C3×S3 C4×S3 D18 S3×C6 C3×M4(2) C8⋊S3 C3×D9 C4×D9 S3×C12 C6×D9 C8⋊D9 C3×C8⋊S3 C12×D9 C3×C8⋊D9 kernel C3×C8⋊D9 C3×C9⋊C8 C3×C72 C12×D9 C8⋊D9 C3×Dic9 C6×D9 C9⋊C8 C72 C4×D9 Dic9 D18 C3×C24 C3×C12 C3×C9 C24 C24 C3×C6 C12 C12 C9 C32 C8 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 3 2 2 3 2 4 4 6 6 4 6 12 8 12 24

Matrix representation of C3×C8⋊D9 in GL2(𝔽73) generated by

 8 0 0 8
,
 22 0 0 51
,
 32 0 0 16
,
 0 16 32 0
G:=sub<GL(2,GF(73))| [8,0,0,8],[22,0,0,51],[32,0,0,16],[0,32,16,0] >;

C3×C8⋊D9 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes D_9
% in TeX

G:=Group("C3xC8:D9");
// GroupNames label

G:=SmallGroup(432,106);
// by ID

G=gap.SmallGroup(432,106);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,80,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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