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G = C4×C9⋊C6  order 216 = 23·33

Direct product of C4 and C9⋊C6

direct product, metacyclic, supersoluble, monomial

Aliases: C4×C9⋊C6, D9⋊C12, C36⋊2C6, D18.C6, Dic9⋊2C6, (C4×D9)⋊C3, C9⋊C12⋊2C2, C9⋊1(C2×C12), C32.(C4×S3), C3.3(S3×C12), C6.13(S3×C6), (C3×C12).8S3, C18.2(C2×C6), (C3×C6).10D6, C12.13(C3×S3), 3- 1+2⋊1(C2×C4), (C4×3- 1+2)⋊2C2, (C2×3- 1+2).2C22, (C2×C9⋊C6).C2, C2.1(C2×C9⋊C6), SmallGroup(216,53)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — C4×C9⋊C6
Chief series
C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C4×C9⋊C6
Lower central
C9 — C4×C9⋊C6
Upper central
C1 — C4

Generators and relations for C4×C9⋊C6
 G = < a,b,c | a4=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >

C1
C2
9C2
9C2
C3
3C3
C4
9C4
9C22
C6
3S3
3C6
3S3
9C6
9C6
C9
C32
2C9
9C2×C4
C12
3D6
3C12
3Dic3
9C2×C6
9C12
C18
D9
D9
C3×C6
2C18
3C3×S3
3C3×S3
3- 1+2
3C4×S3
9C2×C12
C3×C12
C36
Dic9
D18
2C36
3S3×C6
3C3×Dic3
C2×3- 1+2
C9⋊C6
C9⋊C6
C4×D9
3S3×C12
C2×C9⋊C6
C9⋊C12
C4×3- 1+2
C4×C9⋊C6

Smallest permutation representation of C4×C9⋊C6
►On 36 points
Generators in S36
(1 29 11 20)(2 30 12 21)(3 31 13 22)(4 32 14 23)(5 33 15 24)(6 34 16 25)(7 35 17 26)(8 36 18 27)(9 28 10 19)

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

C4×C9⋊C6 is a maximal subgroup of   C72⋊C6  D36⋊6C6  Dic18⋊2C6  D36⋊3C6
C4×C9⋊C6 is a maximal quotient of   C72⋊C6  Dic9⋊C12  D18⋊C12

40 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D6A6B6C6D6E6F6G9A9B9C12A12B12C12D12E12F12G12H12I12J18A18B18C36A···36F
order1222333444466666669991212121212121212121218181836···36
size11992331199233999966622333399996666···6

40 irreducible representations

dim1111111111222222666
type++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6C3×S3C4×S3S3×C6S3×C12C9⋊C6C2×C9⋊C6C4×C9⋊C6
kernelC4×C9⋊C6C9⋊C12C4×3- 1+2C2×C9⋊C6C4×D9C9⋊C6Dic9C36D18D9C3×C12C3×C6C12C32C6C3C4C2C1
# reps1111242228112224112

Matrix representation of C4×C9⋊C6 ►in GL6(𝔽37)

3100000
0310000
0031000
0003100
0000310
0000031
,
0036100
0036000
363636363635
0000136
001001
101001
,
0360000
3600000
111121
0000136
00036360
00360360

G:=sub<GL(6,GF(37))| [31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[0,0,36,0,0,1,0,0,36,0,0,0,36,36,36,0,1,1,1,0,36,0,0,0,0,0,36,1,0,0,0,0,35,36,1,1],[0,36,1,0,0,0,36,0,1,0,0,0,0,0,1,0,0,36,0,0,1,0,36,0,0,0,2,1,36,36,0,0,1,36,0,0] >;
 

C4×C9⋊C6 in GAP, Magma, Sage, TeX 

C_4\times C_9\rtimes C_6
 
% in TeX
 
G:=Group("C4xC9:C6");
 
// GroupNames label
 
G:=SmallGroup(216,53);
 
// by ID
 
G=gap.SmallGroup(216,53);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,3604,736,208,5189]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
 
// generators/relations
 

Export

Subgroup lattice of C4×C9⋊C6 in TeX

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