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## G = S3×Q8⋊2S3order 288 = 25·32

### Direct product of S3 and Q8⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — S3×Q8⋊2S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — S3×Q8⋊2S3
 Lower central C32 — C3×C6 — C3×C12 — S3×Q8⋊2S3
 Upper central C1 — C2 — C4 — Q8

Generators and relations for S3×Q82S3
G = < a,b,c,d,e,f | a3=b2=c4=e3=f2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf=c-1, ce=ec, de=ed, fdf=c-1d, fef=e-1 >

Subgroups: 682 in 146 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×2], S3 [×4], C6 [×2], C6 [×4], C8 [×2], C2×C4 [×2], D4 [×3], Q8, Q8 [×2], C23, C32, Dic3, Dic3, C12 [×2], C12 [×6], D6, D6 [×8], C2×C6 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12 [×4], C3⋊D4, C2×C12 [×2], C3×D4, C3×Q8 [×2], C3×Q8 [×3], C22×S3 [×2], C2×SD16, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S32 [×2], S3×C6, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, C2×C3⋊C8, D4.S3, Q82S3, Q82S3 [×5], C3×SD16, C2×D12, S3×D4, S3×Q8, C6×Q8, C3×C3⋊C8, C324C8, C3⋊D12, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C3×D12, C12⋊S3, Q8×C32, C2×S32, S3×SD16, C2×Q82S3, S3×C3⋊C8, Dic6⋊S3, C325SD16, C3×Q82S3, C3211SD16, S3×D12, C3×S3×Q8, S3×Q82S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C2×SD16, S32, Q82S3 [×2], S3×D4, C2×C3⋊D4, C2×S32, S3×SD16, C2×Q82S3, S3×C3⋊D4, S3×Q82S3

Smallest permutation representation of S3×Q82S3
On 48 points
Generators in S48
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 34)(2 35)(3 36)(4 33)(5 26)(6 27)(7 28)(8 25)(9 30)(10 31)(11 32)(12 29)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 48)(22 45)(23 46)(24 47)
(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 40 7 38)(6 39 8 37)(9 42 11 44)(10 41 12 43)(13 27 15 25)(14 26 16 28)(17 29 19 31)(18 32 20 30)(33 48 35 46)(34 47 36 45)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 47 10)(6 48 11)(7 45 12)(8 46 9)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(1 34)(2 33)(3 36)(4 35)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 47)(22 46)(23 45)(24 48)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 24A 24B order 1 2 2 2 2 2 3 3 3 4 4 4 4 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 12 12 12 24 24 size 1 1 3 3 12 36 2 2 4 2 4 6 12 2 2 4 6 6 24 6 6 18 18 4 4 4 4 8 8 8 8 12 12 12 12 12

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D6 SD16 C3⋊D4 C3⋊D4 S32 Q8⋊2S3 S3×D4 C2×S32 S3×SD16 S3×C3⋊D4 S3×Q8⋊2S3 kernel S3×Q8⋊2S3 S3×C3⋊C8 Dic6⋊S3 C32⋊5SD16 C3×Q8⋊2S3 C32⋊11SD16 S3×D12 C3×S3×Q8 Q8⋊2S3 S3×Q8 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 D12 C3×Q8 C3×S3 Dic3 D6 Q8 S3 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 4 2 2 1 2 1 1 2 2 1

Matrix representation of S3×Q82S3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 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 72 0 0 0 0 0 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 72 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
,
 6 67 0 0 0 0 67 67 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 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 72 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 72 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,67,0,0,0,0,67,67,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

S3×Q82S3 in GAP, Magma, Sage, TeX

S_3\times Q_8\rtimes_2S_3
% in TeX

G:=Group("S3xQ8:2S3");
// GroupNames label

G:=SmallGroup(288,586);
// by ID

G=gap.SmallGroup(288,586);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,100,346,185,80,1356,9414]);
// Polycyclic

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

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