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G = C241D6order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C241D6, Dic68D6, D6.4D12, D12.17D6, Dic3.6D12, C81S32, C3⋊C81D6, C6.3(S3×D4), C8⋊S31S3, C24⋊C21S3, (S3×D12)⋊2C2, (C4×S3).1D6, (S3×C6).1D4, C6.3(C2×D12), C2.8(S3×D12), C325D82C2, C3⋊D243C2, C31(C8⋊D6), C31(Q83D6), (C3×C24)⋊4C22, D6.6D61C2, C323(C8⋊C22), (C3×Dic3).1D4, C12⋊S32C22, C325SD161C2, (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

C1C3×C12 — C241D6
C1C3C32C3×C6C3×C12S3×C12S3×D12 — C241D6
C32C3×C6C3×C12 — C241D6
C1C2C4C8

Generators and relations for C241D6
 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 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×8], C6 [×2], C6 [×3], C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3, Dic3, C12 [×2], C12 [×3], D6, D6 [×11], C2×C6 [×2], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, C4×S3, C4×S3 [×2], D12, D12 [×8], C3⋊D4 [×2], C2×C12, C3×D4, C3×Q8, C22×S3 [×2], C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S32 [×2], S3×C6, S3×C6, C2×C3⋊S3 [×2], C8⋊S3, C24⋊C2, C24⋊C2, D24 [×4], D4⋊S3, Q82S3, C3×M4(2), C3×SD16, C2×D12, C4○D12, S3×D4, Q83S3, C3×C3⋊C8, C3×C24, C6.D6, C3⋊D12 [×2], C3×Dic6, S3×C12, C3×D12, C12⋊S3 [×2], C2×S32, C8⋊D6, Q83D6, C3⋊D24, C325SD16, C3×C8⋊S3, C3×C24⋊C2, C325D8, D6.6D6, S3×D12, C241D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], C8⋊C22, S32, C2×D12, S3×D4, C2×S32, C8⋊D6, Q83D6, S3×D12, C241D6

Smallest permutation representation of C241D6
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 38 17 46 9 30)(2 25 18 33 10 41)(3 36 19 44 11 28)(4 47 20 31 12 39)(5 34 21 42 13 26)(6 45 22 29 14 37)(7 32 23 40 15 48)(8 43 24 27 16 35)
(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 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim111111112222222222244444444
type+++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6D12D12C8⋊C22S32S3×D4C2×S32C8⋊D6Q83D6S3×D12C241D6
kernelC241D6C3⋊D24C325SD16C3×C8⋊S3C3×C24⋊C2C325D8D6.6D6S3×D12C8⋊S3C24⋊C2C3×Dic3S3×C6C3⋊C8C24Dic6C4×S3D12Dic3D6C32C8C6C4C3C3C2C1
# reps111111111111121112211112224

Matrix representation of C241D6 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] >;

C241D6 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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