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## G = S3×D4.S3order 288 = 25·32

### Direct product of S3 and D4.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — S3×D4.S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×Dic6 — S3×D4.S3
 Lower central C32 — C3×C6 — C3×C12 — S3×D4.S3
 Upper central C1 — C2 — C4 — D4

Generators and relations for S3×D4.S3
G = < a,b,c,d,e,f | a3=b2=c4=d2=e3=1, f2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf-1=c-1, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >

Subgroups: 554 in 146 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×S3, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C2×SD16, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, S3×C8, C24⋊C2, C2×C3⋊C8, D4.S3, D4.S3, Q82S3, C3×SD16, C2×Dic6, S3×D4, S3×Q8, C6×D4, C3×C3⋊C8, C324C8, S3×Dic3, C322Q8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C324Q8, D4×C32, S3×C2×C6, S3×SD16, C2×D4.S3, S3×C3⋊C8, Dic6⋊S3, D12.S3, C3×D4.S3, C329SD16, S3×Dic6, C3×S3×D4, S3×D4.S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C2×SD16, S32, D4.S3, S3×D4, C2×C3⋊D4, C2×S32, S3×SD16, C2×D4.S3, S3×C3⋊D4, S3×D4.S3

Smallest permutation representation of S3×D4.S3
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 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 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(22 24)(26 28)(29 31)(33 34)(35 36)(37 38)(39 40)(41 44)(42 43)(46 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 30 25)(22 31 26)(23 32 27)(24 29 28)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(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), (1,2,3,4)(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), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24)(26,28)(29,31)(33,34)(35,36)(37,38)(39,40)(41,44)(42,43)(46,48), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (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), (1,2,3,4)(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), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24)(26,28)(29,31)(33,34)(35,36)(37,38)(39,40)(41,44)(42,43)(46,48), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (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)], [(1,2,3,4),(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)], [(1,4),(2,3),(6,8),(9,11),(13,14),(15,16),(17,20),(18,19),(22,24),(26,28),(29,31),(33,34),(35,36),(37,38),(39,40),(41,44),(42,43),(46,48)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,45,10),(6,46,11),(7,47,12),(8,48,9),(21,30,25),(22,31,26),(23,32,27),(24,29,28),(33,42,37),(34,43,38),(35,44,39),(36,41,40)], [(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)]])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 8A 8B 8C 8D 12A 12B 12C 12D 12E 24A 24B order 1 2 2 2 2 2 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 24 24 size 1 1 3 3 4 12 2 2 4 2 6 12 36 2 2 4 4 4 6 6 8 8 8 12 12 6 6 18 18 4 4 8 12 24 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 D4.S3 S3×D4 C2×S32 S3×SD16 S3×C3⋊D4 S3×D4.S3 kernel S3×D4.S3 S3×C3⋊C8 Dic6⋊S3 D12.S3 C3×D4.S3 C32⋊9SD16 S3×Dic6 C3×S3×D4 D4.S3 S3×D4 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 D12 C3×D4 C3×S3 Dic3 D6 D4 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×D4.S3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 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 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 29 0 0 0 0 10 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
,
 72 29 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 61 28 0 0 0 0 60 12 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 72 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,10,0,0,0,0,29,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,29,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[61,60,0,0,0,0,28,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

S3×D4.S3 in GAP, Magma, Sage, TeX

S_3\times D_4.S_3
% in TeX

G:=Group("S3xD4.S3");
// GroupNames label

G:=SmallGroup(288,576);
// by ID

G=gap.SmallGroup(288,576);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,135,346,185,80,1356,9414]);
// Polycyclic

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

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