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G = C3⋊D36  order 216 = 23·33

The semidirect product of C3 and D36 acting via D36/D18=C2

metabelian, supersoluble, monomial

Aliases: C3⋊2D36, Dic3⋊D9, D18⋊1S3, C18.4D6, C6.4D18, C32.2D12, C6.4S32, (C3×C9)⋊1D4, (C6×D9)⋊1C2, C2.5(S3×D9), C9⋊1(C3⋊D4), (C3×C6).25D6, (C9×Dic3)⋊1C2, (C3×C18).4C22, (C3×Dic3).2S3, C3.1(C3⋊D12), (C2×C9⋊S3)⋊1C2, SmallGroup(216,29)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C18 — C3⋊D36
Chief series
C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — C3⋊D36
Lower central
C3×C9 — C3×C18 — C3⋊D36
Upper central
C1 — C2

Generators and relations for C3⋊D36
 G = < a,b,c | a3=b36=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 390 in 58 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C9, C9, C32, Dic3, C12, D6, C2×C6, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, D12, C3⋊D4, C3×C9, C36, D18, D18, C3×Dic3, S3×C6, C2×C3⋊S3, C3×D9, C9⋊S3, C3×C18, D36, C3⋊D12, C9×Dic3, C6×D9, C2×C9⋊S3, C3⋊D36
Quotients: C1, C2, C22, S3, D4, D6, D9, D12, C3⋊D4, D18, S32, D36, C3⋊D12, S3×D9, C3⋊D36

Smallest permutation representation of C3⋊D36
►On 36 points
Generators in S36
(1 13 25)(2 26 14)(3 15 27)(4 28 16)(5 17 29)(6 30 18)(7 19 31)(8 32 20)(9 21 33)(10 34 22)(11 23 35)(12 36 24)

(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)

(1 9)(2 8)(3 7)(4 6)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)
 
G:=sub<Sym(36)| (1,13,25)(2,26,14)(3,15,27)(4,28,16)(5,17,29)(6,30,18)(7,19,31)(8,32,20)(9,21,33)(10,34,22)(11,23,35)(12,36,24), (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), (1,9)(2,8)(3,7)(4,6)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)>;
 
G:=Group( (1,13,25)(2,26,14)(3,15,27)(4,28,16)(5,17,29)(6,30,18)(7,19,31)(8,32,20)(9,21,33)(10,34,22)(11,23,35)(12,36,24), (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), (1,9)(2,8)(3,7)(4,6)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24) );
 
G=PermutationGroup([[(1,13,25),(2,26,14),(3,15,27),(4,28,16),(5,17,29),(6,30,18),(7,19,31),(8,32,20),(9,21,33),(10,34,22),(11,23,35),(12,36,24)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24)]])
 

C3⋊D36 is a maximal subgroup of   D18.D6  Dic6⋊5D9  D6.D18  S3×D36  D18.3D6  D9×C3⋊D4  D18⋊D6
C3⋊D36 is a maximal quotient of   D36.S3  C6.D36  C3⋊D72  C3⋊Dic36  Dic3⋊Dic9  D18⋊Dic3  C6.18D36

33 conjugacy classes

class 1 2A2B2C3A3B3C 4 6A6B6C6D6E9A9B9C9D9E9F12A12B18A18B18C18D18E18F36A···36F
order1222333466666999999121218181818181836···36
size11185422462241818222444662224446···6

33 irreducible representations

dim111122222222224444
type+++++++++++++++++
imageC1C2C2C2S3S3D4D6D6D9C3⋊D4D12D18D36S32C3⋊D12S3×D9C3⋊D36
kernelC3⋊D36C9×Dic3C6×D9C2×C9⋊S3D18C3×Dic3C3×C9C18C3×C6Dic3C9C32C6C3C6C3C2C1
# reps111111111322361133

Matrix representation of C3⋊D36 ►in GL4(𝔽37) generated by

03600
13600
0010
0001
,
143000
72300
003120
001711
,
0100
1000
00617
001131
G:=sub<GL(4,GF(37))| [0,1,0,0,36,36,0,0,0,0,1,0,0,0,0,1],[14,7,0,0,30,23,0,0,0,0,31,17,0,0,20,11],[0,1,0,0,1,0,0,0,0,0,6,11,0,0,17,31] >;
 

C3⋊D36 in GAP, Magma, Sage, TeX 

C_3\rtimes D_{36}
 
% in TeX
 
G:=Group("C3:D36");
 
// GroupNames label
 
G:=SmallGroup(216,29);
 
// by ID
 
G=gap.SmallGroup(216,29);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,1065,453,1444,2603]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^36=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

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