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G = C3⋊C820D6order 288 = 25·32

9th semidirect product of C3⋊C8 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: C3⋊C820D6, C12.32(C4×S3), C3⋊S32M4(2), C33(S3×M4(2)), C4.Dic36S3, (C2×C12).109D6, C62.48(C2×C4), C326(C2×M4(2)), (C6×C12).69C22, C4.9(C6.D6), C12.31D612C2, C12.29D611C2, C12.146(C22×S3), (C3×C12).147C23, C22.5(C6.D6), C4.93(C2×S32), C6.27(S3×C2×C4), (C2×C4).106S32, (C4×C3⋊S3).5C4, (C3×C3⋊C8)⋊25C22, (C2×C6).19(C4×S3), (C3×C12).59(C2×C4), C2.5(C2×C6.D6), (C4×C3⋊S3).89C22, (C22×C3⋊S3).10C4, C3⋊Dic3.41(C2×C4), (C2×C3⋊Dic3).19C4, (C3×C4.Dic3)⋊12C2, (C3×C6).43(C22×C4), (C2×C4×C3⋊S3).1C2, (C2×C3⋊S3).35(C2×C4), SmallGroup(288,466)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3⋊C820D6
C1C3C32C3×C6C3×C12C3×C3⋊C8C12.29D6 — C3⋊C820D6
C32C3×C6 — C3⋊C820D6
C1C4C2×C4

Generators and relations for C3⋊C820D6
 G = < a,b,c,d | a3=b8=c6=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=dbd=b5, dcd=c-1 >

Subgroups: 594 in 163 conjugacy classes, 54 normal (24 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×10], C6 [×2], C6 [×5], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×6], C12 [×4], C12 [×2], D6 [×14], C2×C6 [×2], C2×C6, C2×C8 [×2], M4(2) [×4], C22×C4, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×4], C24 [×4], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×2], C2×C12, C22×S3 [×3], C2×M4(2), C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×C8 [×4], C8⋊S3 [×4], C4.Dic3 [×2], C3×M4(2) [×2], S3×C2×C4 [×3], C3×C3⋊C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×M4(2) [×2], C12.29D6 [×2], C12.31D6 [×2], C3×C4.Dic3 [×2], C2×C4×C3⋊S3, C3⋊C820D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], M4(2) [×2], C22×C4, C4×S3 [×4], C22×S3 [×2], C2×M4(2), S32, S3×C2×C4 [×2], C6.D6 [×2], C2×S32, S3×M4(2) [×2], C2×C6.D6, C3⋊C820D6

Permutation representations of C3⋊C820D6
On 24 points - transitive group 24T616
Generators in S24
(1 21 9)(2 10 22)(3 23 11)(4 12 24)(5 17 13)(6 14 18)(7 19 15)(8 16 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 9 21)(2 14 22 6 10 18)(3 11 23)(4 16 24 8 12 20)(5 13 17)(7 15 19)
(1 21)(2 18)(3 23)(4 20)(5 17)(6 22)(7 19)(8 24)(10 14)(12 16)

G:=sub<Sym(24)| (1,21,9)(2,10,22)(3,23,11)(4,12,24)(5,17,13)(6,14,18)(7,19,15)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,9,21)(2,14,22,6,10,18)(3,11,23)(4,16,24,8,12,20)(5,13,17)(7,15,19), (1,21)(2,18)(3,23)(4,20)(5,17)(6,22)(7,19)(8,24)(10,14)(12,16)>;

G:=Group( (1,21,9)(2,10,22)(3,23,11)(4,12,24)(5,17,13)(6,14,18)(7,19,15)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,9,21)(2,14,22,6,10,18)(3,11,23)(4,16,24,8,12,20)(5,13,17)(7,15,19), (1,21)(2,18)(3,23)(4,20)(5,17)(6,22)(7,19)(8,24)(10,14)(12,16) );

G=PermutationGroup([(1,21,9),(2,10,22),(3,23,11),(4,12,24),(5,17,13),(6,14,18),(7,19,15),(8,16,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,9,21),(2,14,22,6,10,18),(3,11,23),(4,16,24,8,12,20),(5,13,17),(7,15,19)], [(1,21),(2,18),(3,23),(4,20),(5,17),(6,22),(7,19),(8,24),(10,14),(12,16)])

G:=TransitiveGroup(24,616);

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F6A6B6C···6G8A···8H12A12B12C12D12E···12J24A···24H
order122222333444444666···68···81212121212···1224···24
size11299182241129918224···46···622224···412···12

48 irreducible representations

dim11111111222222444444
type++++++++++++
imageC1C2C2C2C2C4C4C4S3D6D6M4(2)C4×S3C4×S3S32C6.D6C2×S32C6.D6S3×M4(2)C3⋊C820D6
kernelC3⋊C820D6C12.29D6C12.31D6C3×C4.Dic3C2×C4×C3⋊S3C4×C3⋊S3C2×C3⋊Dic3C22×C3⋊S3C4.Dic3C3⋊C8C2×C12C3⋊S3C12C2×C6C2×C4C4C4C22C3C1
# reps12221422242444111144

Matrix representation of C3⋊C820D6 in GL4(𝔽5) generated by

4030
0402
3000
0200
,
0200
4000
0402
2040
,
0030
0402
3010
0200
,
1000
0400
2040
0201
G:=sub<GL(4,GF(5))| [4,0,3,0,0,4,0,2,3,0,0,0,0,2,0,0],[0,4,0,2,2,0,4,0,0,0,0,4,0,0,2,0],[0,0,3,0,0,4,0,2,3,0,1,0,0,2,0,0],[1,0,2,0,0,4,0,2,0,0,4,0,0,0,0,1] >;

C3⋊C820D6 in GAP, Magma, Sage, TeX

C_3\rtimes C_8\rtimes_{20}D_6
% in TeX

G:=Group("C3:C8:20D6");
// GroupNames label

G:=SmallGroup(288,466);
// by ID

G=gap.SmallGroup(288,466);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,422,219,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^6=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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