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G = C24⋊1D6  order 288 = 25·32

1st semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24⋊1D6, Dic6⋊8D6, D6.4D12, D12.17D6, Dic3.6D12, C8⋊1S32, C3⋊C8⋊1D6, C6.3(S3×D4), C8⋊S3⋊1S3, C24⋊C2⋊1S3, (S3×D12)⋊2C2, (C4×S3).1D6, (S3×C6).1D4, C6.3(C2×D12), C2.8(S3×D12), C32⋊5D8⋊2C2, C3⋊D24⋊3C2, C3⋊1(C8⋊D6), C3⋊1(Q8⋊3D6), (C3×C24)⋊4C22, D6.6D6⋊1C2, C32⋊3(C8⋊C22), (C3×Dic3).1D4, C12⋊S3⋊2C22, C32⋊5SD16⋊1C2, (S3×C12).3C22, (C3×C12).42C23, (C3×Dic6)⋊2C22, (C3×D12).2C22, C12.119(C22×S3), C4.42(C2×S32), (C3×C3⋊C8)⋊1C22, (C3×C8⋊S3)⋊3C2, (C3×C24⋊C2)⋊5C2, (C3×C6).26(C2×D4), SmallGroup(288,442)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C24⋊1D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — C24⋊1D6
Lower central
C32 — C3×C6 — C3×C12 — C24⋊1D6
Upper central
C1 — C2 — C4 — C8

Generators and relations for C24⋊1D6
 G = < a,b,c | a24=b6=c2=1, bab-1=a11, cac=a-1, cbc=b-1 >

Subgroups: 834 in 147 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, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2, C24⋊C2, D24, D4⋊S3, Q8⋊2S3, C3×M4(2), C3×SD16, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C3×C3⋊C8, C3×C24, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C12⋊S3, C2×S32, C8⋊D6, Q8⋊3D6, C3⋊D24, C32⋊5SD16, C3×C8⋊S3, C3×C24⋊C2, C32⋊5D8, D6.6D6, S3×D12, C24⋊1D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C8⋊C22, S32, C2×D12, S3×D4, C2×S32, C8⋊D6, Q8⋊3D6, S3×D12, C24⋊1D6

Smallest permutation representation of C24⋊1D6
►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 29 17 37 9 45)(2 40 18 48 10 32)(3 27 19 35 11 43)(4 38 20 46 12 30)(5 25 21 33 13 41)(6 36 22 44 14 28)(7 47 23 31 15 39)(8 34 24 42 16 26)

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

36 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E8A8B12A12B12C12D12E12F12G24A···24H24I24J
order12222233344466666881212121212121224···242424
size116123636224261222412244122244412244···41212

36 irreducible representations

dim111111112222222222244444444
type+++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6D12D12C8⋊C22S32S3×D4C2×S32C8⋊D6Q8⋊3D6S3×D12C24⋊1D6
kernelC24⋊1D6C3⋊D24C32⋊5SD16C3×C8⋊S3C3×C24⋊C2C32⋊5D8D6.6D6S3×D12C8⋊S3C24⋊C2C3×Dic3S3×C6C3⋊C8C24Dic6C4×S3D12Dic3D6C32C8C6C4C3C3C2C1
# reps111111111111121112211112224

Matrix representation of C24⋊1D6 ►in GL8(𝔽73)

720000000
072000000
0059660000
007660000
00000010
00000001
00000100
000072000
,
01000000
7272000000
007140000
007660000
00005023053
00002323530
00000205050
00002005023
,
072000000
720000000
007140000
007660000
00000010
000000072
00001000
000007200

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,59,7,0,0,0,0,0,0,66,66,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,14,66,0,0,0,0,0,0,0,0,50,23,0,20,0,0,0,0,23,23,20,0,0,0,0,0,0,53,50,50,0,0,0,0,53,0,50,23],[0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,14,66,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0] >;
 

C24⋊1D6 in GAP, Magma, Sage, TeX 

C_{24}\rtimes_1D_6
 
% in TeX
 
G:=Group("C24:1D6");
 
// GroupNames label
 
G:=SmallGroup(288,442);
 
// by ID
 
G=gap.SmallGroup(288,442);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,58,346,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^11,c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

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