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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(S3xD4), C8:S3:1S3, C24:C2:1S3, (S3xD12):2C2, (C4xS3).1D6, (S3xC6).1D4, C6.3(C2xD12), C2.8(S3xD12), C32:5D8:2C2, C3:D24:3C2, C3:1(C8:D6), C3:1(Q8:3D6), (C3xC24):4C22, D6.6D6:1C2, C32:3(C8:C22), (C3xDic3).1D4, C12:S3:2C22, C32:5SD16:1C2, (S3xC12).3C22, (C3xC12).42C23, (C3xDic6):2C22, (C3xD12).2C22, C12.119(C22xS3), C4.42(C2xS32), (C3xC3:C8):1C22, (C3xC8:S3):3C2, (C3xC24:C2):5C2, (C3xC6).26(C2xD4), SmallGroup(288,442)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — C24:1D6
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — S3xC12 — S3xD12 — C24:1D6
Lower central
C32 — C3xC6 — C3xC12 — 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, 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, C3:D4, C2xC12, C3xD4, C3xQ8, C22xS3, C8:C22, C3xDic3, C3xDic3, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C8:S3, C24:C2, C24:C2, D24, D4:S3, Q8:2S3, C3xM4(2), C3xSD16, C2xD12, C4oD12, S3xD4, Q8:3S3, C3xC3:C8, C3xC24, C6.D6, C3:D12, C3xDic6, S3xC12, C3xD12, C12:S3, C2xS32, C8:D6, Q8:3D6, C3:D24, C32:5SD16, C3xC8:S3, C3xC24:C2, C32:5D8, D6.6D6, S3xD12, C24:1D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, C8:C22, S32, C2xD12, S3xD4, C2xS32, C8:D6, Q8:3D6, S3xD12, 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:C22S32S3xD4C2xS32C8:D6Q8:3D6S3xD12C24:1D6
kernelC24:1D6C3:D24C32:5SD16C3xC8:S3C3xC24:C2C32:5D8D6.6D6S3xD12C8:S3C24:C2C3xDic3S3xC6C3:C8C24Dic6C4xS3D12Dic3D6C32C8C6C4C3C3C2C1
# 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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