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G = C3×Dic18order 216 = 23·33

Direct product of C3 and Dic18

direct product, metacyclic, supersoluble, monomial

Aliases: C3×Dic18, C36.5C6, C12.7D9, C6.19D18, Dic9.2C6, C32.3Dic6, C4.(C3×D9), (C3×C9)⋊3Q8, C93(C3×Q8), C6.6(S3×C6), C2.3(C6×D9), C12.1(C3×S3), (C3×C36).2C2, C18.9(C2×C6), (C3×C6).45D6, (C3×C12).12S3, C3.1(C3×Dic6), (C3×Dic9).2C2, (C3×C18).13C22, SmallGroup(216,43)

Series: Derived Chief Lower central Upper central

C1C18 — C3×Dic18
C1C3C9C18C3×C18C3×Dic9 — C3×Dic18
C9C18 — C3×Dic18
C1C6C12

Generators and relations for C3×Dic18
 G = < a,b,c | a3=b36=1, c2=b18, ab=ba, ac=ca, cbc-1=b-1 >

2C3
9C4
9C4
2C6
2C9
9Q8
2C12
3Dic3
3Dic3
9C12
9C12
2C18
3Dic6
9C3×Q8
2C36
3C3×Dic3
3C3×Dic3
3C3×Dic6

Smallest permutation representation of C3×Dic18
On 72 points
Generators in S72
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)
(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)
(1 68 19 50)(2 67 20 49)(3 66 21 48)(4 65 22 47)(5 64 23 46)(6 63 24 45)(7 62 25 44)(8 61 26 43)(9 60 27 42)(10 59 28 41)(11 58 29 40)(12 57 30 39)(13 56 31 38)(14 55 32 37)(15 54 33 72)(16 53 34 71)(17 52 35 70)(18 51 36 69)

G:=sub<Sym(72)| (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60), (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), (1,68,19,50)(2,67,20,49)(3,66,21,48)(4,65,22,47)(5,64,23,46)(6,63,24,45)(7,62,25,44)(8,61,26,43)(9,60,27,42)(10,59,28,41)(11,58,29,40)(12,57,30,39)(13,56,31,38)(14,55,32,37)(15,54,33,72)(16,53,34,71)(17,52,35,70)(18,51,36,69)>;

G:=Group( (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60), (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), (1,68,19,50)(2,67,20,49)(3,66,21,48)(4,65,22,47)(5,64,23,46)(6,63,24,45)(7,62,25,44)(8,61,26,43)(9,60,27,42)(10,59,28,41)(11,58,29,40)(12,57,30,39)(13,56,31,38)(14,55,32,37)(15,54,33,72)(16,53,34,71)(17,52,35,70)(18,51,36,69) );

G=PermutationGroup([(1,13,25),(2,14,26),(3,15,27),(4,16,28),(5,17,29),(6,18,30),(7,19,31),(8,20,32),(9,21,33),(10,22,34),(11,23,35),(12,24,36),(37,61,49),(38,62,50),(39,63,51),(40,64,52),(41,65,53),(42,66,54),(43,67,55),(44,68,56),(45,69,57),(46,70,58),(47,71,59),(48,72,60)], [(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)], [(1,68,19,50),(2,67,20,49),(3,66,21,48),(4,65,22,47),(5,64,23,46),(6,63,24,45),(7,62,25,44),(8,61,26,43),(9,60,27,42),(10,59,28,41),(11,58,29,40),(12,57,30,39),(13,56,31,38),(14,55,32,37),(15,54,33,72),(16,53,34,71),(17,52,35,70),(18,51,36,69)])

C3×Dic18 is a maximal subgroup of
C6.D36  C3⋊Dic36  D12.D9  C12.D18  Dic18⋊S3  D12⋊D9  Dic9.D6  C3×Q8×D9

63 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C6A6B6C6D6E9A···9I12A···12H12I12J12K12L18A···18I36A···36R
order1233333444666669···912···121212121218···1836···36
size111122221818112222···22···2181818182···22···2

63 irreducible representations

dim11111122222222222222
type++++-++-+-
imageC1C2C2C3C6C6S3Q8D6D9C3×S3C3×Q8Dic6D18S3×C6C3×D9Dic18C3×Dic6C6×D9C3×Dic18
kernelC3×Dic18C3×Dic9C3×C36Dic18Dic9C36C3×C12C3×C9C3×C6C12C12C9C32C6C6C4C3C3C2C1
# reps121242111322232664612

Matrix representation of C3×Dic18 in GL2(𝔽37) generated by

100
010
,
320
022
,
01
360
G:=sub<GL(2,GF(37))| [10,0,0,10],[32,0,0,22],[0,36,1,0] >;

C3×Dic18 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{18}
% in TeX

G:=Group("C3xDic18");
// GroupNames label

G:=SmallGroup(216,43);
// by ID

G=gap.SmallGroup(216,43);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,3604,208,5189]);
// Polycyclic

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

Export

Subgroup lattice of C3×Dic18 in TeX

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