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## G = C22×C32⋊D6order 432 = 24·33

### Direct product of C22 and C32⋊D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C22×C32⋊D6
 Chief series C1 — C3 — C32 — He3 — C32⋊C6 — C32⋊D6 — C2×C32⋊D6 — C22×C32⋊D6
 Lower central He3 — C22×C32⋊D6
 Upper central C1 — C22

Generators and relations for C22×C32⋊D6
G = < a,b,c,d,e,f | a2=b2=c3=d3=e6=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=fcf=c-1d-1, ede-1=d-1, df=fd, fef=e-1 >

Subgroups: 3107 in 501 conjugacy classes, 109 normal (8 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C32, C32, D6, C2×C6, C2×C6, C24, C3×S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C22×C6, He3, S32, S3×C6, C2×C3⋊S3, C62, C62, S3×C23, C32⋊C6, He3⋊C2, C2×He3, C2×S32, S3×C2×C6, C22×C3⋊S3, C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, C22×He3, C22×S32, C2×C32⋊D6, C22×C32⋊C6, C22×He3⋊C2, C22×C32⋊D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, S32, S3×C23, C2×S32, C32⋊D6, C22×S32, C2×C32⋊D6, C22×C32⋊D6

Smallest permutation representation of C22×C32⋊D6
On 36 points
Generators in S36
(1 10)(2 11)(3 12)(4 9)(5 7)(6 8)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 6)(2 4)(3 5)(7 12)(8 10)(9 11)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(19 34)(20 35)(21 36)(22 31)(23 32)(24 33)
(1 28 25)(2 29 26)(4 18 15)(6 17 14)(8 24 21)(9 19 22)(10 33 36)(11 34 31)
(1 28 25)(2 26 29)(3 30 27)(4 15 18)(5 13 16)(6 17 14)(7 20 23)(8 24 21)(9 22 19)(10 33 36)(11 31 34)(12 35 32)
(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 6)(2 5)(3 4)(7 11)(8 10)(9 12)(13 26)(14 25)(15 30)(16 29)(17 28)(18 27)(19 32)(20 31)(21 36)(22 35)(23 34)(24 33)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2O 3A 3B 3C 3D 6A 6B 6C 6D ··· 6I 6J 6K 6L 6M ··· 6X order 1 2 2 2 2 ··· 2 3 3 3 3 6 6 6 6 ··· 6 6 6 6 6 ··· 6 size 1 1 1 1 9 ··· 9 2 6 6 12 2 2 2 6 ··· 6 12 12 12 18 ··· 18

44 irreducible representations

 dim 1 1 1 1 2 2 2 4 4 6 6 type + + + + + + + + + + + image C1 C2 C2 C2 S3 D6 D6 S32 C2×S32 C32⋊D6 C2×C32⋊D6 kernel C22×C32⋊D6 C2×C32⋊D6 C22×C32⋊C6 C22×He3⋊C2 C22×C3⋊S3 C2×C3⋊S3 C62 C2×C6 C6 C22 C2 # reps 1 12 2 1 2 12 2 1 3 2 6

Matrix representation of C22×C32⋊D6 in GL14(ℤ)

 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 1 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 -1 -1 -1
,
 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -2 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0
,
 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -2 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(14,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[0,-1,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-2,-1,0,0,1,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,-1,-2,0,0,1,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0] >;

C22×C32⋊D6 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes D_6
% in TeX

G:=Group("C2^2xC3^2:D6");
// GroupNames label

G:=SmallGroup(432,545);
// by ID

G=gap.SmallGroup(432,545);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,571,4037,537,14118,7069]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^6=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,e*c*e^-1=f*c*f=c^-1*d^-1,e*d*e^-1=d^-1,d*f=f*d,f*e*f=e^-1>;
// generators/relations

׿
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