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G = C22×3- 1+2  order 108 = 22·33

Direct product of C22 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial

Aliases: C22×3- 1+2, C18⋊2C6, C62.3C3, C3.2C62, C9⋊2(C2×C6), (C2×C18)⋊3C3, C6.5(C3×C6), (C3×C6).4C6, C32.(C2×C6), (C2×C6).7C32, SmallGroup(108,31)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C22×3- 1+2
Chief series
C1 — C3 — C32 — 3- 1+2 — C2×3- 1+2 — C22×3- 1+2
Lower central
C1 — C3 — C22×3- 1+2
Upper central
C1 — C2×C6 — C22×3- 1+2

Generators and relations for C22×3- 1+2
 G = < a,b,c,d | a2=b2=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

C1
C2
C2
C2
C3
3C3
C22
C6
C6
C6
3C6
3C6
3C6
C9
C9
C32
C9
C2×C6
3C2×C6
C3×C6
C3×C6
C18
C18
C18
C3×C6
C18
C18
C18
C18
C18
C18
3- 1+2
C2×C18
C2×C18
C2×C18
C62
C2×3- 1+2
C2×3- 1+2
C2×3- 1+2
C22×3- 1+2

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

(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(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)

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

C22×3- 1+2 is a maximal subgroup of   Dic9⋊C6  C62.C32  3- 1+2⋊A4  C62.6C32  C62.9C32

44 conjugacy classes

class 1 2A2B2C3A3B3C3D6A···6F6G···6L9A···9F18A···18R
order122233336···66···69···918···18
size111111331···13···33···33···3

44 irreducible representations

dim11111133
type++
imageC1C2C3C3C6C63- 1+2C2×3- 1+2
kernelC22×3- 1+2C2×3- 1+2C2×C18C62C18C3×C6C22C2
# reps136218626

Matrix representation of C22×3- 1+2 ►in GL4(𝔽19) generated by

1000
01800
00180
00018
,
18000
0100
0010
0001
,
7000
0010
00011
0100
,
11000
0100
00110
0007
G:=sub<GL(4,GF(19))| [1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,0,0,1,0,1,0,0,0,0,11,0],[11,0,0,0,0,1,0,0,0,0,11,0,0,0,0,7] >;
 

C22×3- 1+2 in GAP, Magma, Sage, TeX 

C_2^2\times 3_-^{1+2}
 
% in TeX
 
G:=Group("C2^2xES-(3,1)");
 
// GroupNames label
 
G:=SmallGroup(108,31);
 
// by ID
 
G=gap.SmallGroup(108,31);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-3,147,253]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^9=d^3=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^4>;
 
// generators/relations
 

Export

Subgroup lattice of C22×3- 1+2 in TeX

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