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G = C4×3- 1+2order 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, C92C12, 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

C1C3 — C4×3- 1+2
C1C3C6C3×C6C2×3- 1+2 — C4×3- 1+2
C1C3 — C4×3- 1+2
C1C12 — 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 >

3C3
3C6
3C12

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⋊C94C6

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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