Copied to
clipboard

G = S3×D4.S3order 288 = 25·32

Direct product of S3 and D4.S3

direct product, metabelian, supersoluble, monomial

Aliases: S3×D4.S3, Dic69D6, D12.6D6, C3⋊C814D6, D4.1S32, (S3×D4).S3, C36(S3×SD16), (S3×Dic6)⋊4C2, (C3×S3)⋊2SD16, (C4×S3).19D6, (C3×D4).20D6, (S3×C6).32D4, C6.148(S3×D4), C329(C2×SD16), (C3×C12).5C23, C12.5(C22×S3), Dic6⋊S38C2, C329SD161C2, D6.20(C3⋊D4), C324C85C22, (C3×Dic3).12D4, (C3×Dic6)⋊6C22, D12.S311C2, C324Q84C22, (S3×C12).12C22, Dic3.4(C3⋊D4), (C3×D12).12C22, (D4×C32).1C22, (S3×C3⋊C8)⋊4C2, C4.5(C2×S32), (C3×S3×D4).1C2, C32(C2×D4.S3), (C3×C3⋊C8)⋊14C22, (C3×D4.S3)⋊5C2, C6.44(C2×C3⋊D4), C2.22(S3×C3⋊D4), (C3×C6).120(C2×D4), SmallGroup(288,576)

Series: Derived Chief Lower central Upper central

C1C3×C12 — S3×D4.S3
C1C3C32C3×C6C3×C12S3×C12S3×Dic6 — S3×D4.S3
C32C3×C6C3×C12 — S3×D4.S3
C1C2C4D4

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 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×2], S3, C6 [×2], C6 [×7], C8 [×2], C2×C4 [×2], D4, D4 [×2], Q8 [×3], C23, C32, Dic3, Dic3 [×4], C12 [×2], C12 [×3], D6, D6 [×3], C2×C6 [×7], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6, Dic6 [×5], C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4 [×2], C3×D4 [×3], C3×Q8, C22×S3, C22×C6, C2×SD16, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6 [×3], C62, S3×C8, C24⋊C2, C2×C3⋊C8, D4.S3, D4.S3 [×5], 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 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C2×SD16, S32, D4.S3 [×2], 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 2A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D6E6F6G6H6I6J6K6L8A8B8C8D12A12B12C12D12E24A24B
order1222223334444666666666666888812121212122424
size11334122242612362244466888121266181844812241212

36 irreducible representations

dim111111112222222222224444448
type++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6SD16C3⋊D4C3⋊D4S32D4.S3S3×D4C2×S32S3×SD16S3×C3⋊D4S3×D4.S3
kernelS3×D4.S3S3×C3⋊C8Dic6⋊S3D12.S3C3×D4.S3C329SD16S3×Dic6C3×S3×D4D4.S3S3×D4C3×Dic3S3×C6C3⋊C8Dic6C4×S3D12C3×D4C3×S3Dic3D6D4S3C6C4C3C2C1
# reps111111111111111124221211221

Matrix representation of S3×D4.S3 in GL6(𝔽73)

100000
010000
0072100
0072000
000010
000001
,
100000
010000
0007200
0072000
000010
000001
,
72290000
1010000
001000
000100
000010
000001
,
72290000
010000
0072000
0007200
000010
000001
,
100000
010000
001000
000100
00007272
000010
,
61280000
60120000
001000
000100
000010
00007272

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

׿
×
𝔽