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G = S3×D24order 288 = 25·32

Direct product of S3 and D24

direct product, metabelian, supersoluble, monomial

Aliases: S3×D24, D127D6, C2416D6, D6.11D12, Dic3.2D12, C84S32, C3⋊C822D6, C31(S3×D8), (S3×C8)⋊1S3, (C3×S3)⋊1D8, C31(C2×D24), C6.2(S3×D4), (S3×C24)⋊4C2, C323(C2×D8), (C3×D24)⋊8C2, (S3×D12)⋊1C2, C6.2(C2×D12), C2.7(S3×D12), C325D86C2, C3⋊D241C2, (C3×C24)⋊6C22, (S3×C6).18D4, (C4×S3).34D6, (C3×D12)⋊1C22, C12⋊S31C22, (C3×C12).41C23, (C3×Dic3).21D4, (S3×C12).42C22, C12.118(C22×S3), C4.41(C2×S32), (C3×C3⋊C8)⋊26C22, (C3×C6).25(C2×D4), SmallGroup(288,441)

Series: Derived Chief Lower central Upper central

C1C3×C12 — S3×D24
C1C3C32C3×C6C3×C12S3×C12S3×D12 — S3×D24
C32C3×C6C3×C12 — S3×D24
C1C2C4C8

Generators and relations for S3×D24
 G = < a,b,c,d | a3=b2=c24=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 994 in 163 conjugacy classes, 44 normal (30 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×2], S3 [×8], C6 [×2], C6 [×5], C8, C8, C2×C4, D4 [×6], C23 [×2], C32, Dic3, C12 [×2], C12 [×2], D6, D6 [×16], C2×C6 [×3], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×2], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, D12 [×2], D12 [×8], C3⋊D4 [×2], C2×C12, C3×D4 [×2], C22×S3 [×4], C2×D8, C3×Dic3, C3×C12, S32 [×4], S3×C6, S3×C6 [×2], C2×C3⋊S3 [×2], S3×C8, D24, D24 [×5], D4⋊S3 [×2], C2×C24, C3×D8, C2×D12 [×2], S3×D4 [×2], C3×C3⋊C8, C3×C24, C3⋊D12 [×2], S3×C12, C3×D12 [×2], C12⋊S3 [×2], C2×S32 [×2], C2×D24, S3×D8, C3⋊D24 [×2], S3×C24, C3×D24, C325D8, S3×D12 [×2], S3×D24
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], D8 [×2], C2×D4, D12 [×2], C22×S3 [×2], C2×D8, S32, D24 [×2], C2×D12, S3×D4, C2×S32, C2×D24, S3×D8, S3×D12, S3×D24

Smallest permutation representation of S3×D24
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 28)(2 27)(3 26)(4 25)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F12G24A24B24C24D24E···24J24K24L24M24N
order122222223334466666668888121212121212122424242424···2424242424
size113312123636224262246624242266224446622224···46666

45 irreducible representations

dim111111222222222222444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6D6D8D12D12D24S32S3×D4C2×S32S3×D8S3×D12S3×D24
kernelS3×D24C3⋊D24S3×C24C3×D24C325D8S3×D12S3×C8D24C3×Dic3S3×C6C3⋊C8C24C4×S3D12C3×S3Dic3D6S3C8C6C4C3C2C1
# reps121112111112124228111224

Matrix representation of S3×D24 in GL6(𝔽73)

100000
010000
0072100
0072000
000010
000001
,
100000
010000
0007200
0072000
000010
000001
,
0320000
57410000
001000
000100
0000072
0000172
,
72710000
010000
0072000
0007200
0000172
0000072

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],[0,57,0,0,0,0,32,41,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72] >;

S3×D24 in GAP, Magma, Sage, TeX

S_3\times D_{24}
% in TeX

G:=Group("S3xD24");
// GroupNames label

G:=SmallGroup(288,441);
// by ID

G=gap.SmallGroup(288,441);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,142,346,80,1356,9414]);
// 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^-1>;
// generators/relations

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