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G = C22×3- 1+2order 108 = 22·33

Direct product of C22 and 3- 1+2

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

Aliases: C22×3- 1+2, C182C6, C62.3C3, C3.2C62, C92(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

C1C3 — C22×3- 1+2
C1C3C323- 1+2C2×3- 1+2 — C22×3- 1+2
C1C3 — C22×3- 1+2
C1C2×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 >

3C3
3C6
3C6
3C6
3C2×C6

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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