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## G = S3×C32⋊2Q8order 432 = 24·33

### Direct product of S3 and C32⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — S3×C32⋊2Q8
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — S3×C32⋊2Q8
 Lower central C33 — C32×C6 — S3×C32⋊2Q8
 Upper central C1 — C2

Generators and relations for S3×C322Q8
G = < a,b,c,d,e,f | a3=b2=c3=d3=e4=1, f2=e2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=c-1, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >

Subgroups: 980 in 198 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2×C4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C2×Q8, C3×S3, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C2×Dic6, S3×Q8, S3×C32, C32×C6, S3×Dic3, S3×Dic3, C322Q8, C322Q8, C3×Dic6, S3×C12, C6×Dic3, C324Q8, C2×C3⋊Dic3, C32×Dic3, C3×C3⋊Dic3, C3×C3⋊Dic3, C335C4, S3×C3×C6, S3×Dic6, C2×C322Q8, C3×S3×Dic3, C3×C322Q8, S3×C3⋊Dic3, C334Q8, C335Q8, S3×C322Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, S3×Q8, C322Q8, C2×S32, S3×Dic6, C2×C322Q8, S33, S3×C322Q8

Smallest permutation representation of S3×C322Q8
On 48 points
Generators in S48
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 45 10)(6 46 11)(7 47 12)(8 48 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 34)(2 35)(3 36)(4 33)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 14 19)(2 20 15)(3 16 17)(4 18 13)(5 10 45)(6 46 11)(7 12 47)(8 48 9)(21 25 30)(22 31 26)(23 27 32)(24 29 28)(33 42 37)(34 38 43)(35 44 39)(36 40 41)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(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 24 3 22)(2 23 4 21)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 30 15 32)(14 29 16 31)(17 26 19 28)(18 25 20 27)(33 45 35 47)(34 48 36 46)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O order 1 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 size 1 1 3 3 2 2 2 4 4 4 8 6 6 18 18 18 54 2 2 2 4 4 4 6 6 6 6 8 12 12 6 6 6 6 12 ··· 12 18 18 18 18 36

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8 type + + + + + + + + - + + + - + + - - + - + - image C1 C2 C2 C2 C2 C2 S3 S3 Q8 D6 D6 D6 Dic6 S32 S32 S3×Q8 C32⋊2Q8 C2×S32 S3×Dic6 S33 S3×C32⋊2Q8 kernel S3×C32⋊2Q8 C3×S3×Dic3 C3×C32⋊2Q8 S3×C3⋊Dic3 C33⋊4Q8 C33⋊5Q8 S3×Dic3 C32⋊2Q8 S3×C32 C3×Dic3 C3⋊Dic3 S3×C6 C3×S3 Dic3 D6 C32 S3 C6 C3 C2 C1 # reps 1 2 1 1 2 1 2 1 2 4 3 2 8 2 1 1 2 3 4 1 1

Matrix representation of S3×C322Q8 in GL6(𝔽13)

 12 1 0 0 0 0 12 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
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 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 12 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 3 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 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×C322Q8 in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("S3xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(432,603);
// by ID

G=gap.SmallGroup(432,603);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,298,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^3=d^3=e^4=1,f^2=e^2,b*a*b=a^-1,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,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations

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