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## G = C18×Dic6order 432 = 24·33

### Direct product of C18 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C18×Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — C9×Dic3 — Dic3×C18 — C18×Dic6
 Lower central C3 — C6 — C18×Dic6
 Upper central C1 — C2×C18 — C2×C36

Generators and relations for C18×Dic6
G = < a,b,c | a18=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 212 in 130 conjugacy classes, 81 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C18, C18, C18, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C2×Dic6, C6×Q8, C3×C18, C3×C18, C2×C36, C2×C36, Q8×C9, C3×Dic6, C6×Dic3, C6×C12, C9×Dic3, C3×C36, C6×C18, Q8×C18, C6×Dic6, C9×Dic6, Dic3×C18, C6×C36, C18×Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, C9, D6, C2×C6, C2×Q8, C18, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, C2×C18, S3×C6, C2×Dic6, C6×Q8, S3×C9, Q8×C9, C22×C18, C3×Dic6, S3×C2×C6, S3×C18, Q8×C18, C6×Dic6, C9×Dic6, S3×C2×C18, C18×Dic6

Smallest permutation representation of C18×Dic6
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)
(1 136 106 120 13 130 100 114 7 142 94 126)(2 137 107 121 14 131 101 115 8 143 95 109)(3 138 108 122 15 132 102 116 9 144 96 110)(4 139 91 123 16 133 103 117 10 127 97 111)(5 140 92 124 17 134 104 118 11 128 98 112)(6 141 93 125 18 135 105 119 12 129 99 113)(19 79 64 52 25 85 70 40 31 73 58 46)(20 80 65 53 26 86 71 41 32 74 59 47)(21 81 66 54 27 87 72 42 33 75 60 48)(22 82 67 37 28 88 55 43 34 76 61 49)(23 83 68 38 29 89 56 44 35 77 62 50)(24 84 69 39 30 90 57 45 36 78 63 51)
(1 66 100 33)(2 67 101 34)(3 68 102 35)(4 69 103 36)(5 70 104 19)(6 71 105 20)(7 72 106 21)(8 55 107 22)(9 56 108 23)(10 57 91 24)(11 58 92 25)(12 59 93 26)(13 60 94 27)(14 61 95 28)(15 62 96 29)(16 63 97 30)(17 64 98 31)(18 65 99 32)(37 131 76 109)(38 132 77 110)(39 133 78 111)(40 134 79 112)(41 135 80 113)(42 136 81 114)(43 137 82 115)(44 138 83 116)(45 139 84 117)(46 140 85 118)(47 141 86 119)(48 142 87 120)(49 143 88 121)(50 144 89 122)(51 127 90 123)(52 128 73 124)(53 129 74 125)(54 130 75 126)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 9A ··· 9F 9G ··· 9L 12A ··· 12P 12Q ··· 12X 18A ··· 18R 18S ··· 18AJ 36A ··· 36AJ 36AK ··· 36BH order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 1 1 2 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

162 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 type + + + + + - + + - image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 S3 Q8 D6 D6 C3×S3 Dic6 C3×Q8 S3×C6 S3×C6 S3×C9 Q8×C9 C3×Dic6 S3×C18 S3×C18 C9×Dic6 kernel C18×Dic6 C9×Dic6 Dic3×C18 C6×C36 C6×Dic6 C3×Dic6 C6×Dic3 C6×C12 C2×Dic6 Dic6 C2×Dic3 C2×C12 C2×C36 C3×C18 C36 C2×C18 C2×C12 C18 C3×C6 C12 C2×C6 C2×C4 C6 C6 C4 C22 C2 # reps 1 4 2 1 2 8 4 2 6 24 12 6 1 2 2 1 2 4 4 4 2 6 12 8 12 6 24

Matrix representation of C18×Dic6 in GL5(𝔽37)

 4 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 36 0 0 0 0 0 11 0 0 0 0 0 27 0 0 0 0 0 20 20 0 0 0 4 17
,
 36 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 15 23 0 0 0 32 22

G:=sub<GL(5,GF(37))| [4,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[36,0,0,0,0,0,11,0,0,0,0,0,27,0,0,0,0,0,20,4,0,0,0,20,17],[36,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,15,32,0,0,0,23,22] >;

C18×Dic6 in GAP, Magma, Sage, TeX

C_{18}\times {\rm Dic}_6
% in TeX

G:=Group("C18xDic6");
// GroupNames label

G:=SmallGroup(432,341);
// by ID

G=gap.SmallGroup(432,341);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,590,142,192,14118]);
// Polycyclic

G:=Group<a,b,c|a^18=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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