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G = C3×Dic18  order 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, C9⋊3(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

Derived series
C1 — C18 — C3×Dic18
Chief series
C1 — C3 — C9 — C18 — C3×C18 — C3×Dic9 — C3×Dic18
Lower central
C9 — C18 — C3×Dic18
Upper central
C1 — C6 — C12

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

C1
C2
C3
C3
2C3
C4
9C4
9C4
C6
C6
2C6
C32
C9
2C9
9Q8
C12
C12
2C12
3Dic3
3Dic3
9C12
9C12
C18
C3×C6
2C18
C3×C9
3Dic6
9C3×Q8
C36
Dic9
Dic9
C3×C12
2C36
3C3×Dic3
3C3×Dic3
C3×C18
Dic18
3C3×Dic6
C3×C36
C3×Dic9
C3×Dic9
C3×Dic18

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

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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