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## G = C18×C3⋊C8order 432 = 24·33

### Direct product of C18 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C18×C3⋊C8
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — C9×C3⋊C8 — C18×C3⋊C8
 Lower central C3 — C18×C3⋊C8
 Upper central C1 — C2×C36

Generators and relations for C18×C3⋊C8
G = < a,b,c | a18=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 116 in 82 conjugacy classes, 57 normal (39 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, C3×C9, C36, C36, C2×C18, C2×C18, C3×C12, C62, C2×C3⋊C8, C2×C24, C3×C18, C3×C18, C72, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C3×C36, C6×C18, C2×C72, C6×C3⋊C8, C9×C3⋊C8, C6×C36, C18×C3⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C9, Dic3, C12, D6, C2×C6, C2×C8, C18, C3×S3, C3⋊C8, C24, C2×Dic3, C2×C12, C36, C2×C18, C3×Dic3, S3×C6, C2×C3⋊C8, C2×C24, S3×C9, C72, C2×C36, C3×C3⋊C8, C6×Dic3, C9×Dic3, S3×C18, C2×C72, C6×C3⋊C8, C9×C3⋊C8, Dic3×C18, C18×C3⋊C8

Smallest permutation representation of C18×C3⋊C8
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 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 67 61)(56 68 62)(57 69 63)(58 70 64)(59 71 65)(60 72 66)(73 79 85)(74 80 86)(75 81 87)(76 82 88)(77 83 89)(78 84 90)(91 103 97)(92 104 98)(93 105 99)(94 106 100)(95 107 101)(96 108 102)(109 115 121)(110 116 122)(111 117 123)(112 118 124)(113 119 125)(114 120 126)(127 139 133)(128 140 134)(129 141 135)(130 142 136)(131 143 137)(132 144 138)
(1 100 21 46 124 130 76 62)(2 101 22 47 125 131 77 63)(3 102 23 48 126 132 78 64)(4 103 24 49 109 133 79 65)(5 104 25 50 110 134 80 66)(6 105 26 51 111 135 81 67)(7 106 27 52 112 136 82 68)(8 107 28 53 113 137 83 69)(9 108 29 54 114 138 84 70)(10 91 30 37 115 139 85 71)(11 92 31 38 116 140 86 72)(12 93 32 39 117 141 87 55)(13 94 33 40 118 142 88 56)(14 95 34 41 119 143 89 57)(15 96 35 42 120 144 90 58)(16 97 36 43 121 127 73 59)(17 98 19 44 122 128 74 60)(18 99 20 45 123 129 75 61)

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

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

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

216 conjugacy classes

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

216 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 C3 C4 C4 C6 C6 C8 C9 C12 C12 C18 C18 C24 C36 C36 C72 S3 Dic3 D6 Dic3 C3×S3 C3⋊C8 C3×Dic3 S3×C6 C3×Dic3 S3×C9 C3×C3⋊C8 C9×Dic3 S3×C18 C9×Dic3 C9×C3⋊C8 kernel C18×C3⋊C8 C9×C3⋊C8 C6×C36 C6×C3⋊C8 C3×C36 C6×C18 C3×C3⋊C8 C6×C12 C3×C18 C2×C3⋊C8 C3×C12 C62 C3⋊C8 C2×C12 C3×C6 C12 C2×C6 C6 C2×C36 C36 C36 C2×C18 C2×C12 C18 C12 C12 C2×C6 C2×C4 C6 C4 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 6 4 4 12 6 16 12 12 48 1 1 1 1 2 4 2 2 2 6 8 6 6 6 24

Matrix representation of C18×C3⋊C8 in GL4(𝔽73) generated by

 16 0 0 0 0 18 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 8
,
 10 0 0 0 0 72 0 0 0 0 0 1 0 0 72 0
G:=sub<GL(4,GF(73))| [16,0,0,0,0,18,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,8],[10,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0] >;

C18×C3⋊C8 in GAP, Magma, Sage, TeX

C_{18}\times C_3\rtimes C_8
% in TeX

G:=Group("C18xC3:C8");
// GroupNames label

G:=SmallGroup(432,126);
// by ID

G=gap.SmallGroup(432,126);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,84,142,192,14118]);
// Polycyclic

G:=Group<a,b,c|a^18=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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