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## G = C18.Dic6order 432 = 24·33

### 2nd non-split extension by C18 of Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C18.Dic6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C6×C18 — Dic3×C18 — C18.Dic6
 Lower central C3×C9 — C3×C18 — C18.Dic6
 Upper central C1 — C22

Generators and relations for C18.Dic6
G = < a,b,c | a6=b36=1, c2=b18, bab-1=a-1, ac=ca, cbc-1=a3b-1 >

Subgroups: 444 in 94 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C3×C18, C2×Dic9, C2×Dic9, C2×C36, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×C18, Dic9⋊C4, C62.C22, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, C18.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, D9, Dic6, C4×S3, C3⋊D4, D18, S32, Dic3⋊C4, Dic18, C4×D9, C9⋊D4, C6.D6, D6⋊S3, C322Q8, S3×D9, Dic9⋊C4, C62.C22, C9⋊Dic6, C18.D6, D6⋊D9, C18.Dic6

Smallest permutation representation of C18.Dic6
On 144 points
Generators in S144
(1 47 13 59 25 71)(2 72 26 60 14 48)(3 49 15 61 27 37)(4 38 28 62 16 50)(5 51 17 63 29 39)(6 40 30 64 18 52)(7 53 19 65 31 41)(8 42 32 66 20 54)(9 55 21 67 33 43)(10 44 34 68 22 56)(11 57 23 69 35 45)(12 46 36 70 24 58)(73 119 85 131 97 143)(74 144 98 132 86 120)(75 121 87 133 99 109)(76 110 100 134 88 122)(77 123 89 135 101 111)(78 112 102 136 90 124)(79 125 91 137 103 113)(80 114 104 138 92 126)(81 127 93 139 105 115)(82 116 106 140 94 128)(83 129 95 141 107 117)(84 118 108 142 96 130)
(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)
(1 132 19 114)(2 73 20 91)(3 130 21 112)(4 107 22 89)(5 128 23 110)(6 105 24 87)(7 126 25 144)(8 103 26 85)(9 124 27 142)(10 101 28 83)(11 122 29 140)(12 99 30 81)(13 120 31 138)(14 97 32 79)(15 118 33 136)(16 95 34 77)(17 116 35 134)(18 93 36 75)(37 96 55 78)(38 117 56 135)(39 94 57 76)(40 115 58 133)(41 92 59 74)(42 113 60 131)(43 90 61 108)(44 111 62 129)(45 88 63 106)(46 109 64 127)(47 86 65 104)(48 143 66 125)(49 84 67 102)(50 141 68 123)(51 82 69 100)(52 139 70 121)(53 80 71 98)(54 137 72 119)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18I 18J ··· 18R 36A ··· 36L order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 4 6 6 18 18 54 54 2 ··· 2 4 4 4 2 2 2 4 4 4 6 6 6 6 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + - + + + - - + - + + - - + - + - image C1 C2 C2 C2 C4 S3 S3 D4 Q8 D6 D6 D9 Dic6 C4×S3 C3⋊D4 Dic6 C4×S3 C3⋊D4 D18 Dic18 C4×D9 C9⋊D4 S32 C6.D6 D6⋊S3 C32⋊2Q8 S3×D9 C9⋊Dic6 C18.D6 D6⋊D9 kernel C18.Dic6 C6×Dic9 Dic3×C18 C2×C9⋊Dic3 C9⋊Dic3 C2×Dic9 C6×Dic3 C3×C18 C3×C18 C2×C18 C62 C2×Dic3 C18 C18 C18 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 1 1 1 1 1 3 2 2 2 2 2 2 3 6 6 6 1 1 1 1 3 3 3 3

Matrix representation of C18.Dic6 in GL6(𝔽37)

 0 1 0 0 0 0 36 36 0 0 0 0 0 0 0 36 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 24 26 0 0 0 0 2 13 0 0 0 0 0 0 33 8 0 0 0 0 12 4 0 0 0 0 0 0 11 6 0 0 0 0 31 17
,
 32 27 0 0 0 0 10 5 0 0 0 0 0 0 30 23 0 0 0 0 14 7 0 0 0 0 0 0 1 0 0 0 0 0 1 36

G:=sub<GL(6,GF(37))| [0,36,0,0,0,0,1,36,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[24,2,0,0,0,0,26,13,0,0,0,0,0,0,33,12,0,0,0,0,8,4,0,0,0,0,0,0,11,31,0,0,0,0,6,17],[32,10,0,0,0,0,27,5,0,0,0,0,0,0,30,14,0,0,0,0,23,7,0,0,0,0,0,0,1,1,0,0,0,0,0,36] >;

C18.Dic6 in GAP, Magma, Sage, TeX

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

G:=Group("C18.Dic6");
// GroupNames label

G:=SmallGroup(432,89);
// by ID

G=gap.SmallGroup(432,89);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,36,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c|a^6=b^36=1,c^2=b^18,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;
// generators/relations

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