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G = C4×3- 1+2  order 108 = 22·33

Direct product of C4 and 3- 1+2

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C4×3- 1+2, C36⋊C3, C9⋊2C12, C18.2C6, C32.C12, C12.2C32, (C3×C12).C3, C6.3(C3×C6), (C3×C6).3C6, C3.2(C3×C12), C2.(C2×3- 1+2), C12○(C2×3- 1+2), (C2×3- 1+2).2C2, SmallGroup(108,14)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C4×3- 1+2
Chief series
C1 — C3 — C6 — C3×C6 — C2×3- 1+2 — C4×3- 1+2
Lower central
C1 — C3 — C4×3- 1+2
Upper central
C1 — C12 — C4×3- 1+2

Generators and relations for C4×3- 1+2
 G = < a,b,c | a4=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

C1
C2
C3
3C3
C4
C6
3C6
C32
C9
C9
C9
C12
3C12
C18
C18
C3×C6
C18
3- 1+2
C36
C3×C12
C36
C36
C2×3- 1+2
C4×3- 1+2

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

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

C4×3- 1+2 is a maximal subgroup of   C9⋊C24  C36.C6  D36⋊C3  C36.A4  Q8⋊C9⋊4C6

44 conjugacy classes

class 1  2 3A3B3C3D4A4B6A6B6C6D9A···9F12A12B12C12D12E12F12G12H18A···18F36A···36L
order1233334466669···9121212121212121218···1836···36
size1111331111333···3111133333···33···3

44 irreducible representations

dim111111111333
type++
imageC1C2C3C3C4C6C6C12C123- 1+2C2×3- 1+2C4×3- 1+2
kernelC4×3- 1+2C2×3- 1+2C36C3×C123- 1+2C18C3×C6C9C32C4C2C1
# reps1162262124224

Matrix representation of C4×3- 1+2 ►in GL3(𝔽13) generated by

800
080
008
,
008
300
020
,
300
090
001
G:=sub<GL(3,GF(13))| [8,0,0,0,8,0,0,0,8],[0,3,0,0,0,2,8,0,0],[3,0,0,0,9,0,0,0,1] >;
 

C4×3- 1+2 in GAP, Magma, Sage, TeX 

C_4\times 3_-^{1+2}
 
% in TeX
 
G:=Group("C4xES-(3,1)");
 
// GroupNames label
 
G:=SmallGroup(108,14);
 
// by ID
 
G=gap.SmallGroup(108,14);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-2,-3,90,186,322]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
 
// generators/relations
 

Export

Subgroup lattice of C4×3- 1+2 in TeX

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