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## G = C32×A4order 108 = 22·33

### Direct product of C32 and A4

Aliases: C32×A4, C22⋊C33, C623C3, (C2×C6)⋊C32, SmallGroup(108,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×A4
 Chief series C1 — C22 — A4 — C3×A4 — C32×A4
 Lower central C22 — C32×A4
 Upper central C1 — C32

Generators and relations for C32×A4
G = < a,b,c,d,e | a3=b3=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 140 in 62 conjugacy classes, 34 normal (5 characteristic)
C1, C2, C3 [×4], C3 [×9], C22, C6 [×4], C32, C32 [×12], A4 [×9], C2×C6 [×4], C3×C6, C33, C3×A4 [×12], C62, C32×A4
Quotients: C1, C3 [×13], C32 [×13], A4, C33, C3×A4 [×4], C32×A4

Smallest permutation representation of C32×A4
On 36 points
Generators in S36
(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 23 28)(2 24 29)(3 22 30)(4 31 26)(5 32 27)(6 33 25)(7 15 35)(8 13 36)(9 14 34)(10 21 18)(11 19 16)(12 20 17)
(1 16)(2 17)(3 18)(4 9)(5 7)(6 8)(10 22)(11 23)(12 24)(13 33)(14 31)(15 32)(19 28)(20 29)(21 30)(25 36)(26 34)(27 35)
(1 15)(2 13)(3 14)(4 21)(5 19)(6 20)(7 28)(8 29)(9 30)(10 26)(11 27)(12 25)(16 32)(17 33)(18 31)(22 34)(23 35)(24 36)
(1 3 2)(4 20 7)(5 21 8)(6 19 9)(10 36 27)(11 34 25)(12 35 26)(13 32 18)(14 33 16)(15 31 17)(22 24 23)(28 30 29)

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

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

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

C32×A4 is a maximal subgroup of   C324S4  C62.16C32  He32A4  C62.6C32
C32×A4 is a maximal quotient of   C62.25C32  He3.2A4  C62.9C32

36 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Z 6A ··· 6H order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 3 1 ··· 1 4 ··· 4 3 ··· 3

36 irreducible representations

 dim 1 1 1 3 3 type + + image C1 C3 C3 A4 C3×A4 kernel C32×A4 C3×A4 C62 C32 C3 # reps 1 24 2 1 8

Matrix representation of C32×A4 in GL4(𝔽7) generated by

 4 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 1 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 1 0 0 0 0 0 0 1 0 6 6 6 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 1 0 0 0 6 6 6
,
 1 0 0 0 0 4 0 0 0 0 0 4 0 3 3 3
G:=sub<GL(4,GF(7))| [4,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,0,6,1,0,0,6,0,0,1,6,0],[1,0,0,0,0,0,1,6,0,1,0,6,0,0,0,6],[1,0,0,0,0,4,0,3,0,0,0,3,0,0,4,3] >;

C32×A4 in GAP, Magma, Sage, TeX

C_3^2\times A_4
% in TeX

G:=Group("C3^2xA4");
// GroupNames label

G:=SmallGroup(108,41);
// by ID

G=gap.SmallGroup(108,41);
# by ID

G:=PCGroup([5,-3,-3,-3,-2,2,1083,2029]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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