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## G = C22×C32⋊4D6order 432 = 24·33

### Direct product of C22 and C32⋊4D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C22×C32⋊4D6
 Chief series C1 — C3 — C32 — C33 — C3×C3⋊S3 — C32⋊4D6 — C2×C32⋊4D6 — C22×C32⋊4D6
 Lower central C33 — C22×C32⋊4D6
 Upper central C1 — C22

Generators and relations for C22×C324D6
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=c-1, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 3448 in 642 conjugacy classes, 135 normal (5 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, C33, S32, S3×C6, C2×C3⋊S3, C62, C62, S3×C23, C3×C3⋊S3, C32×C6, C2×S32, S3×C2×C6, C22×C3⋊S3, C324D6, C6×C3⋊S3, C3×C62, C22×S32, C2×C324D6, C2×C6×C3⋊S3, C22×C324D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, S32, S3×C23, C2×S32, C324D6, C22×S32, C2×C324D6, C22×C324D6

Smallest permutation representation of C22×C324D6
On 48 points
Generators in S48
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 35)(14 36)(15 31)(16 32)(17 33)(18 34)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)(25 39)(26 40)(27 41)(28 42)(29 37)(30 38)
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 48)(26 43)(27 44)(28 45)(29 46)(30 47)
(1 5 3)(2 4 6)(7 11 9)(8 10 12)(13 17 15)(14 16 18)(19 21 23)(20 24 22)(25 27 29)(26 30 28)(31 35 33)(32 34 36)(37 39 41)(38 42 40)(43 47 45)(44 46 48)
(1 3 5)(2 6 4)(7 9 11)(8 12 10)(13 15 17)(14 18 16)(19 21 23)(20 24 22)(25 27 29)(26 30 28)(31 33 35)(32 36 34)(37 39 41)(38 42 40)(43 47 45)(44 46 48)
(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)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 21)(8 20)(9 19)(10 24)(11 23)(12 22)(13 37)(14 42)(15 41)(16 40)(17 39)(18 38)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)

G:=sub<Sym(48)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,35)(14,36)(15,31)(16,32)(17,33)(18,34)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(25,39)(26,40)(27,41)(28,42)(29,37)(30,38), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,48)(26,43)(27,44)(28,45)(29,46)(30,47), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18)(19,21,23)(20,24,22)(25,27,29)(26,30,28)(31,35,33)(32,34,36)(37,39,41)(38,42,40)(43,47,45)(44,46,48), (1,3,5)(2,6,4)(7,9,11)(8,12,10)(13,15,17)(14,18,16)(19,21,23)(20,24,22)(25,27,29)(26,30,28)(31,33,35)(32,36,34)(37,39,41)(38,42,40)(43,47,45)(44,46,48), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,21)(8,20)(9,19)(10,24)(11,23)(12,22)(13,37)(14,42)(15,41)(16,40)(17,39)(18,38)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)>;

G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,35)(14,36)(15,31)(16,32)(17,33)(18,34)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(25,39)(26,40)(27,41)(28,42)(29,37)(30,38), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,48)(26,43)(27,44)(28,45)(29,46)(30,47), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18)(19,21,23)(20,24,22)(25,27,29)(26,30,28)(31,35,33)(32,34,36)(37,39,41)(38,42,40)(43,47,45)(44,46,48), (1,3,5)(2,6,4)(7,9,11)(8,12,10)(13,15,17)(14,18,16)(19,21,23)(20,24,22)(25,27,29)(26,30,28)(31,33,35)(32,36,34)(37,39,41)(38,42,40)(43,47,45)(44,46,48), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,21)(8,20)(9,19)(10,24)(11,23)(12,22)(13,37)(14,42)(15,41)(16,40)(17,39)(18,38)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34) );

G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,35),(14,36),(15,31),(16,32),(17,33),(18,34),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45),(25,39),(26,40),(27,41),(28,42),(29,37),(30,38)], [(1,33),(2,34),(3,35),(4,36),(5,31),(6,32),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,48),(26,43),(27,44),(28,45),(29,46),(30,47)], [(1,5,3),(2,4,6),(7,11,9),(8,10,12),(13,17,15),(14,16,18),(19,21,23),(20,24,22),(25,27,29),(26,30,28),(31,35,33),(32,34,36),(37,39,41),(38,42,40),(43,47,45),(44,46,48)], [(1,3,5),(2,6,4),(7,9,11),(8,12,10),(13,15,17),(14,18,16),(19,21,23),(20,24,22),(25,27,29),(26,30,28),(31,33,35),(32,36,34),(37,39,41),(38,42,40),(43,47,45),(44,46,48)], [(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),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,21),(8,20),(9,19),(10,24),(11,23),(12,22),(13,37),(14,42),(15,41),(16,40),(17,39),(18,38),(25,33),(26,32),(27,31),(28,36),(29,35),(30,34)]])

60 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2O 3A 3B 3C 3D ··· 3H 6A ··· 6I 6J ··· 6X 6Y ··· 6AJ order 1 2 2 2 2 ··· 2 3 3 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 1 9 ··· 9 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 18 ··· 18

60 irreducible representations

 dim 1 1 1 2 2 2 4 4 4 4 type + + + + + + + + image C1 C2 C2 S3 D6 D6 S32 C2×S32 C32⋊4D6 C2×C32⋊4D6 kernel C22×C32⋊4D6 C2×C32⋊4D6 C2×C6×C3⋊S3 C22×C3⋊S3 C2×C3⋊S3 C62 C2×C6 C6 C22 C2 # reps 1 12 3 3 18 3 3 9 2 6

Matrix representation of C22×C324D6 in GL6(ℤ)

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

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C22×C324D6 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes_4D_6
% in TeX

G:=Group("C2^2xC3^2:4D6");
// GroupNames label

G:=SmallGroup(432,769);
// by ID

G=gap.SmallGroup(432,769);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,1124,571,2028,14118]);
// 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=c^-1,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

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