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G = C9×C8⋊S3order 432 = 24·33

Direct product of C9 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×C8⋊S3
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — S3×C36 — C9×C8⋊S3
 Lower central C3 — C6 — C9×C8⋊S3
 Upper central C1 — C36 — C72

Generators and relations for C9×C8⋊S3
G = < a,b,c,d | a9=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 124 in 68 conjugacy classes, 39 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), C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C3×M4(2), S3×C9, C3×C18, C72, C72, C2×C36, C3×C3⋊C8, C3×C24, S3×C12, C9×Dic3, C3×C36, S3×C18, C9×M4(2), C3×C8⋊S3, C9×C3⋊C8, C3×C72, S3×C36, C9×C8⋊S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C9, C12, D6, C2×C6, M4(2), C18, C3×S3, C4×S3, C2×C12, C36, C2×C18, S3×C6, C8⋊S3, C3×M4(2), S3×C9, C2×C36, S3×C12, S3×C18, C9×M4(2), C3×C8⋊S3, S3×C36, C9×C8⋊S3

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

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

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

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

162 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 ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E ··· 12J 12K 12L 18A ··· 18F 18G ··· 18L 18M ··· 18R 24A ··· 24P 24Q 24R 24S 24T 36A ··· 36L 36M ··· 36X 36Y ··· 36AD 72A ··· 72AJ 72AK ··· 72AV 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 9 ··· 9 12 12 12 12 12 ··· 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 24 ··· 24 24 24 24 24 36 ··· 36 36 ··· 36 36 ··· 36 72 ··· 72 72 ··· 72 size 1 1 6 1 1 2 2 2 1 1 6 1 1 2 2 2 6 6 2 2 6 6 1 ··· 1 2 ··· 2 1 1 1 1 2 ··· 2 6 6 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 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 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C9 C12 C12 C18 C18 C18 C36 C36 S3 D6 M4(2) C3×S3 C4×S3 S3×C6 C8⋊S3 C3×M4(2) S3×C9 S3×C12 S3×C18 C9×M4(2) C3×C8⋊S3 S3×C36 C9×C8⋊S3 kernel C9×C8⋊S3 C9×C3⋊C8 C3×C72 S3×C36 C3×C8⋊S3 C9×Dic3 S3×C18 C3×C3⋊C8 C3×C24 S3×C12 C8⋊S3 C3×Dic3 S3×C6 C3⋊C8 C24 C4×S3 Dic3 D6 C72 C36 C3×C9 C24 C18 C12 C9 C32 C8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 6 4 4 6 6 6 12 12 1 1 2 2 2 2 4 4 6 4 6 12 8 12 24

Matrix representation of C9×C8⋊S3 in GL2(𝔽73) generated by

 2 0 0 2
,
 10 0 0 63
,
 64 0 0 8
,
 0 1 1 0
G:=sub<GL(2,GF(73))| [2,0,0,2],[10,0,0,63],[64,0,0,8],[0,1,1,0] >;

C9×C8⋊S3 in GAP, Magma, Sage, TeX

C_9\times C_8\rtimes S_3
% in TeX

G:=Group("C9xC8:S3");
// GroupNames label

G:=SmallGroup(432,110);
// by ID

G=gap.SmallGroup(432,110);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,1037,92,142,192,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^8=c^3=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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