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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 [×3], C2 [×12], C3, C3 [×3], C22, C22 [×34], S3 [×28], C6 [×3], C6 [×21], C23 [×15], C32 [×2], C32, D6 [×90], C2×C6, C2×C6 [×21], C24, C3×S3 [×20], C3⋊S3 [×8], C3×C6 [×6], C3×C6 [×3], C22×S3 [×43], C22×C6 [×3], He3, S32 [×32], S3×C6 [×30], C2×C3⋊S3 [×12], C62 [×2], C62, S3×C23 [×3], C32⋊C6 [×8], He3⋊C2 [×4], C2×He3 [×3], C2×S32 [×24], S3×C2×C6 [×5], C22×C3⋊S3 [×2], C32⋊D6 [×16], C2×C32⋊C6 [×12], C2×He3⋊C2 [×6], C22×He3, C22×S32 [×2], C2×C32⋊D6 [×12], C22×C32⋊C6 [×2], C22×He3⋊C2, C22×C32⋊D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], S32, S3×C23 [×2], C2×S32 [×3], C32⋊D6, C22×S32, C2×C32⋊D6 [×3], C22×C32⋊D6

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

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

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

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

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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