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## G = S32×C2×C6order 432 = 24·33

### Direct product of C2×C6, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C2×C6
 Chief series C1 — C3 — C32 — C33 — S3×C32 — C3×S32 — S32×C6 — S32×C2×C6
 Lower central C32 — S32×C2×C6
 Upper central C1 — C2×C6

Generators and relations for S32×C2×C6
G = < a,b,c,d,e,f | a2=b6=c3=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2080 in 642 conjugacy classes, 208 normal (12 characteristic)
C1, C2, C2, C3, C3, C3, C22, C22, S3, S3, C6, C6, C23, C32, C32, C32, D6, D6, C2×C6, C2×C6, C2×C6, C24, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C22×S3, C22×C6, C33, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, C62, C62, S3×C23, C23×C6, S3×C32, C3×C3⋊S3, C32×C6, C2×S32, S3×C2×C6, S3×C2×C6, C22×C3⋊S3, C2×C62, C3×S32, S3×C3×C6, C6×C3⋊S3, C3×C62, C22×S32, S3×C22×C6, S32×C6, S3×C62, C2×C6×C3⋊S3, S32×C2×C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, S32, S3×C6, S3×C23, C23×C6, C2×S32, S3×C2×C6, C3×S32, C22×S32, S3×C22×C6, S32×C6, S32×C2×C6

Smallest permutation representation of S32×C2×C6
On 48 points
Generators in S48
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 28)(8 29)(9 30)(10 25)(11 26)(12 27)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 40)(20 41)(21 42)(22 37)(23 38)(24 39)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 3A 3B 3C ··· 3H 3I 3J 3K 6A ··· 6F 6G ··· 6X 6Y ··· 6AN 6AO ··· 6AW 6AX ··· 6BU 6BV ··· 6CC order 1 2 2 2 2 ··· 2 2 2 2 2 3 3 3 ··· 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 1 3 ··· 3 9 9 9 9 1 1 2 ··· 2 4 4 4 1 ··· 1 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 9 ··· 9

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D6 D6 C3×S3 S3×C6 S3×C6 S32 C2×S32 C3×S32 S32×C6 kernel S32×C2×C6 S32×C6 S3×C62 C2×C6×C3⋊S3 C22×S32 C2×S32 S3×C2×C6 C22×C3⋊S3 S3×C2×C6 S3×C6 C62 C22×S3 D6 C2×C6 C2×C6 C6 C22 C2 # reps 1 12 2 1 2 24 4 2 2 12 2 4 24 4 1 3 2 6

Matrix representation of S32×C2×C6 in GL6(𝔽7)

 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 6 0 0 0 0 0 0 6
,
 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 6 1 0 0 0 0 6 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 0 6 0 0 0 0 6 0
,
 6 1 0 0 0 0 6 0 0 0 0 0 0 0 6 1 0 0 0 0 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 6 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 6 0 0 0 0 0 0 0 6 0 0 0 0 0 0 6

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

S32×C2×C6 in GAP, Magma, Sage, TeX

S_3^2\times C_2\times C_6
% in TeX

G:=Group("S3^2xC2xC6");
// GroupNames label

G:=SmallGroup(432,767);
// by ID

G=gap.SmallGroup(432,767);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,2028,14118]);
// Polycyclic

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

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