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## G = C22×C32⋊A4order 432 = 24·33

### Direct product of C22 and C32⋊A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C22×C32⋊A4
 Chief series C1 — C22 — C2×C6 — C62 — C32⋊A4 — C2×C32⋊A4 — C22×C32⋊A4
 Lower central C22 — C2×C6 — C22×C32⋊A4
 Upper central C1 — C2×C6 — C62

Generators and relations for C22×C32⋊A4
G = < a,b,c,d,e,f,g | a2=b2=c3=d3=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg-1=cd-1, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 727 in 224 conjugacy classes, 50 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C23, C32, C32, A4, C2×C6, C2×C6, C24, C3×C6, C3×C6, C2×A4, C22×C6, C22×C6, He3, C3×A4, C62, C62, C22×A4, C23×C6, C23×C6, C2×He3, C6×A4, C2×C62, C2×C62, C32⋊A4, C22×He3, A4×C2×C6, C22×C62, C2×C32⋊A4, C22×C32⋊A4
Quotients: C1, C2, C3, C22, C6, C32, A4, C2×C6, C3×C6, C2×A4, He3, C3×A4, C62, C22×A4, C2×He3, C6×A4, C32⋊A4, C22×He3, A4×C2×C6, C2×C32⋊A4, C22×C32⋊A4

Smallest permutation representation of C22×C32⋊A4
On 36 points
Generators in S36
(1 11)(2 10)(3 12)(4 7)(5 9)(6 8)(13 36)(14 34)(15 35)(16 32)(17 33)(18 31)(19 25)(20 26)(21 27)(22 30)(23 28)(24 29)
(1 10)(2 11)(3 7)(4 12)(5 8)(6 9)(13 21)(14 19)(15 20)(16 28)(17 29)(18 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(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 5 3)(2 6 4)(7 10 8)(9 12 11)(13 14 15)(16 18 17)(19 20 21)(22 24 23)(25 26 27)(28 30 29)(31 33 32)(34 35 36)
(1 2)(3 4)(5 6)(7 12)(8 9)(10 11)(16 23)(17 24)(18 22)(28 32)(29 33)(30 31)
(13 27)(14 25)(15 26)(16 23)(17 24)(18 22)(19 34)(20 35)(21 36)(28 32)(29 33)(30 31)
(1 17 19)(2 24 34)(3 18 21)(4 22 36)(5 16 20)(6 23 35)(7 30 13)(8 28 15)(9 32 26)(10 29 14)(11 33 25)(12 31 27)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E ··· 3J 6A ··· 6F 6G ··· 6AR 6AS ··· 6BJ order 1 2 2 2 2 2 2 2 3 3 3 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 1 3 3 3 3 1 1 3 3 12 ··· 12 1 ··· 1 3 ··· 3 12 ··· 12

80 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 A4 C2×A4 He3 C3×A4 C2×He3 C6×A4 C32⋊A4 C2×C32⋊A4 kernel C22×C32⋊A4 C2×C32⋊A4 A4×C2×C6 C22×C62 C6×A4 C2×C62 C62 C3×C6 C24 C2×C6 C23 C6 C22 C2 # reps 1 3 6 2 18 6 1 3 2 2 6 6 6 18

Matrix representation of C22×C32⋊A4 in GL4(𝔽7) generated by

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

C22×C32⋊A4 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes A_4
% in TeX

G:=Group("C2^2xC3^2:A4");
// GroupNames label

G:=SmallGroup(432,550);
// by ID

G=gap.SmallGroup(432,550);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,353,2287,3989]);
// Polycyclic

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

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