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G = D24:S3  order 288 = 25·32

2nd semidirect product of D24 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C24:2D6, D12:8D6, D24:2S3, D6.5D12, Dic3.7D12, C8:2S32, C3:C8:2D6, C6.4(S3xD4), C8:S3:3S3, (S3xD12):3C2, (C3xD24):2C2, (S3xC6).2D4, (C4xS3).2D6, C2.9(S3xD12), C6.4(C2xD12), C24:2S3:1C2, C3:D24:4C2, C3:1(D8:S3), C3:2(C8:D6), (C3xC24):2C22, D12:5S3:1C2, (C3xD12):2C22, C32:4(C8:C22), (C3xDic3).2D4, D12.S3:1C2, (S3xC12).4C22, (C3xC12).43C23, C32:4Q8:2C22, C12:S3.2C22, C12.120(C22xS3), C4.43(C2xS32), (C3xC3:C8):2C22, (C3xC8:S3):1C2, (C3xC6).27(C2xD4), SmallGroup(288,443)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — D24:S3
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — S3xC12 — S3xD12 — D24:S3
Lower central
C32 — C3xC6 — C3xC12 — D24:S3
Upper central
C1 — C2 — C4 — C8

Generators and relations for D24:S3
 G = < a,b,c,d | a24=b2=c3=d2=1, bab=a-1, ac=ca, dad=a13, bc=cb, dbd=a12b, dcd=c-1 >

Subgroups: 778 in 146 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3xS3, C3:S3, C3xC6, C3:C8, C24, C24, Dic6, C4xS3, C4xS3, D12, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C8:C22, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C8:S3, C24:C2, D24, D24, D4:S3, D4.S3, C3xM4(2), C3xD8, C2xD12, C4oD12, S3xD4, D4:2S3, C3xC3:C8, C3xC24, S3xDic3, D6:S3, C3:D12, S3xC12, C3xD12, C32:4Q8, C12:S3, C2xS32, C8:D6, D8:S3, C3:D24, D12.S3, C3xC8:S3, C3xD24, C24:2S3, D12:5S3, S3xD12, D24:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, C8:C22, S32, C2xD12, S3xD4, C2xS32, C8:D6, D8:S3, S3xD12, D24:S3

Smallest permutation representation of D24:S3
►On 48 points
Generators in S48
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 48)(22 47)(23 46)(24 45)

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

36 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F8A8B12A12B12C12D12E12F24A···24H24I24J
order1222223334446666668812121212121224···242424
size116121236224263622412242441222444124···41212

36 irreducible representations

dim11111111222222222244444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D12D12C8:C22S32S3xD4C2xS32C8:D6D8:S3S3xD12D24:S3
kernelD24:S3C3:D24D12.S3C3xC8:S3C3xD24C24:2S3D12:5S3S3xD12C8:S3D24C3xDic3S3xC6C3:C8C24C4xS3D12Dic3D6C32C8C6C4C3C3C2C1
# reps11111111111112122211112224

Matrix representation of D24:S3 ►in GL8(F73)

720000000
072000000
005970000
0066660000
00000010
00000001
000007200
00001000
,
720000000
072000000
0066660000
005970000
000066320
0000667041
0000320676
000004166
,
01000000
7272000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
7272000000
00100000
00010000
0000320676
00000326767
00006767410
0000667041

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,59,66,0,0,0,0,0,0,7,66,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,66,59,0,0,0,0,0,0,66,7,0,0,0,0,0,0,0,0,6,6,32,0,0,0,0,0,6,67,0,41,0,0,0,0,32,0,67,6,0,0,0,0,0,41,6,6],[0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,32,0,67,6,0,0,0,0,0,32,67,67,0,0,0,0,67,67,41,0,0,0,0,0,6,67,0,41] >;
 

D24:S3 in GAP, Magma, Sage, TeX 

D_{24}\rtimes S_3
 
% in TeX
 
G:=Group("D24:S3");
 
// GroupNames label
 
G:=SmallGroup(288,443);
 
// by ID
 
G=gap.SmallGroup(288,443);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,142,346,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^24=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^13,b*c=c*b,d*b*d=a^12*b,d*c*d=c^-1>;
 
// generators/relations
 

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