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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(S3×D4), C8⋊S3⋊3S3, (S3×D12)⋊3C2, (C3×D24)⋊2C2, (S3×C6).2D4, (C4×S3).2D6, C2.9(S3×D12), C6.4(C2×D12), C24⋊2S3⋊1C2, C3⋊D24⋊4C2, C3⋊1(D8⋊S3), C3⋊2(C8⋊D6), (C3×C24)⋊2C22, D12⋊5S3⋊1C2, (C3×D12)⋊2C22, C32⋊4(C8⋊C22), (C3×Dic3).2D4, D12.S3⋊1C2, (S3×C12).4C22, (C3×C12).43C23, C32⋊4Q8⋊2C22, C12⋊S3.2C22, C12.120(C22×S3), C4.43(C2×S32), (C3×C3⋊C8)⋊2C22, (C3×C8⋊S3)⋊1C2, (C3×C6).27(C2×D4), SmallGroup(288,443)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D24⋊S3
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — D24⋊S3
Lower central
C32 — C3×C6 — C3×C12 — 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, C2×C4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2, D24, D24, D4⋊S3, D4.S3, C3×M4(2), C3×D8, C2×D12, C4○D12, S3×D4, D4⋊2S3, C3×C3⋊C8, C3×C24, S3×Dic3, D6⋊S3, C3⋊D12, S3×C12, C3×D12, C32⋊4Q8, C12⋊S3, C2×S32, C8⋊D6, D8⋊S3, C3⋊D24, D12.S3, C3×C8⋊S3, C3×D24, C24⋊2S3, D12⋊5S3, S3×D12, D24⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C8⋊C22, S32, C2×D12, S3×D4, C2×S32, C8⋊D6, D8⋊S3, S3×D12, 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⋊C22S32S3×D4C2×S32C8⋊D6D8⋊S3S3×D12D24⋊S3
kernelD24⋊S3C3⋊D24D12.S3C3×C8⋊S3C3×D24C24⋊2S3D12⋊5S3S3×D12C8⋊S3D24C3×Dic3S3×C6C3⋊C8C24C4×S3D12Dic3D6C32C8C6C4C3C3C2C1
# 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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