Copied to
clipboard

G = S3×C24⋊C2order 288 = 25·32

Direct product of S3 and C24⋊C2

direct product, metabelian, supersoluble, monomial

Aliases: S3×C24⋊C2, C2417D6, Dic67D6, D12.16D6, D6.10D12, Dic3.1D12, C85S32, C3⋊C821D6, (S3×C8)⋊4S3, C6.1(S3×D4), (S3×C24)⋊9C2, C31(S3×SD16), C2.6(S3×D12), C6.1(C2×D12), C242S37C2, (S3×Dic6)⋊1C2, (C3×S3)⋊1SD16, (S3×C6).17D4, (S3×D12).1C2, (C4×S3).33D6, C324(C2×SD16), (C3×C24)⋊14C22, C325SD165C2, D12.S35C2, C12.66(C22×S3), (C3×C12).40C23, (C3×Dic3).20D4, (C3×Dic6)⋊1C22, (C3×D12).1C22, C324Q81C22, (S3×C12).41C22, C12⋊S3.1C22, C4.40(C2×S32), C31(C2×C24⋊C2), (C3×C24⋊C2)⋊9C2, (C3×C3⋊C8)⋊28C22, (C3×C6).24(C2×D4), SmallGroup(288,440)

Series: Derived Chief Lower central Upper central

C1C3×C12 — S3×C24⋊C2
C1C3C32C3×C6C3×C12S3×C12S3×D12 — S3×C24⋊C2
C32C3×C6C3×C12 — S3×C24⋊C2
C1C2C4C8

Generators and relations for S3×C24⋊C2
 G = < a,b,c,d | a3=b2=c24=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >

Subgroups: 746 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, C8, C2×C4 [×2], D4 [×3], Q8 [×3], C23, C32, Dic3, Dic3 [×4], C12 [×2], C12 [×3], 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, C24 [×2], C24 [×2], Dic6, Dic6 [×5], C4×S3, C4×S3, D12, D12 [×4], C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3 [×2], C2×SD16, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32 [×2], S3×C6, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, C24⋊C2 [×5], D4.S3, Q82S3, C2×C24, C3×SD16, C2×Dic6, C2×D12, S3×D4, S3×Q8, C3×C3⋊C8, C3×C24, S3×Dic3, C3⋊D12, C322Q8, C3×Dic6, S3×C12, C3×D12, C324Q8, C12⋊S3, C2×S32, C2×C24⋊C2, S3×SD16, D12.S3, C325SD16, S3×C24, C3×C24⋊C2, C242S3, S3×Dic6, S3×D12, S3×C24⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, D12 [×2], C22×S3 [×2], C2×SD16, S32, C24⋊C2 [×2], C2×D12, S3×D4, C2×S32, C2×C24⋊C2, S3×SD16, S3×D12, S3×C24⋊C2

Smallest permutation representation of S3×C24⋊C2
On 48 points
Generators in S48
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)
(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 38)(2 25)(3 36)(4 47)(5 34)(6 45)(7 32)(8 43)(9 30)(10 41)(11 28)(12 39)(13 26)(14 37)(15 48)(16 35)(17 46)(18 33)(19 44)(20 31)(21 42)(22 29)(23 40)(24 27)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D6E6F8A8B8C8D12A12B12C12D12E12F12G12H24A24B24C24D24E···24J24K24L24M24N
order1222223334444666666888812121212121212122424242424···2424242424
size113312362242612362246624226622444662422224···46666

45 irreducible representations

dim111111112222222222222444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6SD16D12D12C24⋊C2S32S3×D4C2×S32S3×SD16S3×D12S3×C24⋊C2
kernelS3×C24⋊C2D12.S3C325SD16S3×C24C3×C24⋊C2C242S3S3×Dic6S3×D12S3×C8C24⋊C2C3×Dic3S3×C6C3⋊C8C24Dic6C4×S3D12C3×S3Dic3D6S3C8C6C4C3C2C1
# reps111111111111121114228111224

Matrix representation of S3×C24⋊C2 in GL6(𝔽73)

100000
010000
000100
00727200
000010
000001
,
100000
010000
0072000
001100
000010
000001
,
6760000
67670000
0072000
0007200
00007272
000010
,
100000
0720000
0072000
0007200
000010
00007272

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

S3×C24⋊C2 in GAP, Magma, Sage, TeX

S_3\times C_{24}\rtimes C_2
% in TeX

G:=Group("S3xC24:C2");
// GroupNames label

G:=SmallGroup(288,440);
// by ID

G=gap.SmallGroup(288,440);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,58,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^24=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^11>;
// generators/relations

׿
×
𝔽