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G = C22×C9⋊C6  order 216 = 23·33

Direct product of C22 and C9⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C22×C9⋊C6, D18⋊3C6, C62.8S3, 3- 1+2⋊C23, C18⋊(C2×C6), D9⋊(C2×C6), C9⋊(C22×C6), (C2×C18)⋊4C6, C6.23(S3×C6), (C3×C6).21D6, (C22×D9)⋊3C3, C32.(C22×S3), (C2×3- 1+2)⋊C22, (C22×3- 1+2)⋊2C2, C3.3(S3×C2×C6), (C2×C6).18(C3×S3), SmallGroup(216,111)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — C22×C9⋊C6
Chief series
C1 — C3 — C9 — 3- 1+2 — C9⋊C6 — C2×C9⋊C6 — C22×C9⋊C6
Lower central
C9 — C22×C9⋊C6
Upper central
C1 — C22

Generators and relations for C22×C9⋊C6
 G = < a,b,c,d | a2=b2=c9=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 336 in 101 conjugacy classes, 47 normal (12 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C9, C9, C32, D6, C2×C6, C2×C6, D9, C18, C18, C3×S3, C3×C6, C22×S3, C22×C6, 3- 1+2, D18, C2×C18, C2×C18, S3×C6, C62, C9⋊C6, C2×3- 1+2, C22×D9, S3×C2×C6, C2×C9⋊C6, C22×3- 1+2, C22×C9⋊C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C3×S3, C22×S3, C22×C6, S3×C6, C9⋊C6, S3×C2×C6, C2×C9⋊C6, C22×C9⋊C6

Smallest permutation representation of C22×C9⋊C6
►On 36 points
Generators in S36
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)

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

C22×C9⋊C6 is a maximal subgroup of   D18⋊C12
C22×C9⋊C6 is a maximal quotient of   D36⋊6C6  Dic18⋊2C6  D36⋊3C6

40 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C6A6B6C6D···6I6J···6Q9A9B9C18A···18I
order122222223336666···66···699918···18
size111199992332223···39···96666···6

40 irreducible representations

dim111111222266
type+++++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6C9⋊C6C2×C9⋊C6
kernelC22×C9⋊C6C2×C9⋊C6C22×3- 1+2C22×D9D18C2×C18C62C3×C6C2×C6C6C22C2
# reps1612122132613

Matrix representation of C22×C9⋊C6 ►in GL8(𝔽19)

10000000
01000000
001800000
000180000
000018000
000001800
000000180
000000018
,
180000000
018000000
001800000
000180000
000018000
000001800
000000180
000000018
,
181000000
180000000
0011181700
000011800
000001801
00110181818
000001800
001001800
,
012000000
120000000
00010000
00100000
00000010
0011001818
0011181800
00000100

G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18],[18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18],[18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,18,1,0,0,0,0,0,0,17,18,18,18,18,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,1,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0] >;
 

C22×C9⋊C6 in GAP, Magma, Sage, TeX 

C_2^2\times C_9\rtimes C_6
 
% in TeX
 
G:=Group("C2^2xC9:C6");
 
// GroupNames label
 
G:=SmallGroup(216,111);
 
// by ID
 
G=gap.SmallGroup(216,111);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,3604,382,208,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^9=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^2>;
 
// generators/relations
 

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