Copied to
clipboard

G = C335(C2×C8)  order 432 = 24·33

2nd semidirect product of C33 and C2×C8 acting via C2×C8/C2=C2×C4

metabelian, soluble, monomial, A-group

Aliases: C335(C2×C8), C326(S3×C8), C334C85C2, C322C85S3, C33⋊C22C8, C3⋊Dic3.24D6, Dic3.2(C32⋊C4), (C32×Dic3).1C4, C2.3(S3×C32⋊C4), C6.5(C2×C32⋊C4), (C3×C6).30(C4×S3), C31(C3⋊S33C8), C338(C2×C4).3C2, (C3×C322C8)⋊5C2, (C32×C6).5(C2×C4), (C2×C33⋊C2).1C4, (C3×C3⋊Dic3).27C22, SmallGroup(432,571)

Series: Derived Chief Lower central Upper central

C1C33 — C335(C2×C8)
C1C3C33C32×C6C3×C3⋊Dic3C338(C2×C4) — C335(C2×C8)
C33 — C335(C2×C8)
C1C2

Generators and relations for C335(C2×C8)
 G = < a,b,c,d,e | a3=b3=c3=d2=e8=1, ab=ba, ac=ca, dad=a-1, eae-1=ab-1, bc=cb, dbd=b-1, ebe-1=a-1b-1, dcd=c-1, ce=ec, de=ed >

Subgroups: 864 in 96 conjugacy classes, 20 normal (18 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×2], C22, S3 [×10], C6, C6 [×4], C8 [×2], C2×C4, C32, C32 [×4], Dic3, Dic3 [×2], C12 [×3], D6 [×5], C2×C8, C3⋊S3 [×10], C3×C6, C3×C6 [×4], C3⋊C8, C24, C4×S3 [×3], C33, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×5], S3×C8, C33⋊C2 [×2], C32×C6, C322C8, C322C8, C6.D6 [×2], C4×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, C3⋊S33C8, C3×C322C8, C334C8, C338(C2×C4), C335(C2×C8)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D6, C2×C8, C4×S3, C32⋊C4, S3×C8, C2×C32⋊C4, C3⋊S33C8, S3×C32⋊C4, C335(C2×C8)

Permutation representations of C335(C2×C8)
On 24 points - transitive group 24T1308
Generators in S24
(1 12 21)(2 13 22)(3 23 14)(4 24 15)(5 16 17)(6 9 18)(7 19 10)(8 20 11)
(2 22 13)(4 15 24)(6 18 9)(8 11 20)
(1 21 12)(2 22 13)(3 23 14)(4 24 15)(5 17 16)(6 18 9)(7 19 10)(8 20 11)
(1 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,1308);

36 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D12E12F24A24B24C24D
order1222333334444666668888888812121212121224242424
size1127272448833992448899992727272712121212181818181818

36 irreducible representations

dim1111111222244488
type++++++++++
imageC1C2C2C2C4C4C8S3D6C4×S3S3×C8C32⋊C4C2×C32⋊C4C3⋊S33C8S3×C32⋊C4C335(C2×C8)
kernelC335(C2×C8)C3×C322C8C334C8C338(C2×C4)C32×Dic3C2×C33⋊C2C33⋊C2C322C8C3⋊Dic3C3×C6C32Dic3C6C3C2C1
# reps1111228112422422

Matrix representation of C335(C2×C8) in GL6(𝔽73)

100000
010000
000001
000100
001000
000010
,
100000
010000
000101
0072717272
001100
000110
,
0720000
1720000
001000
000100
000010
000001
,
010000
100000
00727200
000100
00072072
00072720
,
4600000
0460000
001010100
00006310
0000063
00063063

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,1,71,1,1,0,0,0,72,0,1,0,0,1,72,0,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,72,1,72,72,0,0,0,0,0,72,0,0,0,0,72,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,10,0,0,0,0,0,10,0,0,63,0,0,10,63,0,0,0,0,0,10,63,63] >;

C335(C2×C8) in GAP, Magma, Sage, TeX

C_3^3\rtimes_5(C_2\times C_8)
% in TeX

G:=Group("C3^3:5(C2xC8)");
// GroupNames label

G:=SmallGroup(432,571);
// by ID

G=gap.SmallGroup(432,571);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,36,58,1411,298,1356,1027,14118]);
// Polycyclic

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

׿
×
𝔽