G = C33⋊5(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: C33⋊5(C2×C8), C32⋊6(S3×C8), C33⋊4C8⋊5C2, C32⋊2C8⋊5S3, C33⋊C2⋊2C8, 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), C3⋊1(C3⋊S3⋊3C8), C33⋊8(C2×C4).3C2, (C3×C32⋊2C8)⋊5C2, (C32×C6).5(C2×C4), (C2×C33⋊C2).1C4, (C3×C3⋊Dic3).27C22, SmallGroup(432,571)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊8(C2×C4) — C33⋊5(C2×C8) |
| C33 — C33⋊5(C2×C8) |
Generators and relations for C33⋊5(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, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, Dic3, C12, D6, C2×C8, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C33, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×C8, C33⋊C2, C32×C6, C32⋊2C8, C32⋊2C8, C6.D6, C4×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, C3⋊S3⋊3C8, C3×C32⋊2C8, C33⋊4C8, C33⋊8(C2×C4), C33⋊5(C2×C8)
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C4×S3, C32⋊C4, S3×C8, C2×C32⋊C4, C3⋊S3⋊3C8, S3×C32⋊C4, C33⋊5(C2×C8)
►On 24 points - transitive group 24T1308
(1 9 21)(2 10 22)(3 23 11)(4 24 12)(5 13 17)(6 14 18)(7 19 15)(8 20 16)
(2 22 10)(4 12 24)(6 18 14)(8 16 20)
(1 21 9)(2 22 10)(3 23 11)(4 24 12)(5 17 13)(6 18 14)(7 19 15)(8 20 16)
(1 5)(2 6)(3 7)(4 8)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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)| (1,9,21)(2,10,22)(3,23,11)(4,24,12)(5,13,17)(6,14,18)(7,19,15)(8,20,16), (2,22,10)(4,12,24)(6,18,14)(8,16,20), (1,21,9)(2,22,10)(3,23,11)(4,24,12)(5,17,13)(6,18,14)(7,19,15)(8,20,16), (1,5)(2,6)(3,7)(4,8)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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( (1,9,21)(2,10,22)(3,23,11)(4,24,12)(5,13,17)(6,14,18)(7,19,15)(8,20,16), (2,22,10)(4,12,24)(6,18,14)(8,16,20), (1,21,9)(2,22,10)(3,23,11)(4,24,12)(5,17,13)(6,18,14)(7,19,15)(8,20,16), (1,5)(2,6)(3,7)(4,8)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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([[(1,9,21),(2,10,22),(3,23,11),(4,24,12),(5,13,17),(6,14,18),(7,19,15),(8,20,16)], [(2,22,10),(4,12,24),(6,18,14),(8,16,20)], [(1,21,9),(2,22,10),(3,23,11),(4,24,12),(5,17,13),(6,18,14),(7,19,15),(8,20,16)], [(1,5),(2,6),(3,7),(4,8),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(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 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 27 | 27 | 2 | 4 | 4 | 8 | 8 | 3 | 3 | 9 | 9 | 2 | 4 | 4 | 8 | 8 | 9 | 9 | 9 | 9 | 27 | 27 | 27 | 27 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | C4×S3 | S3×C8 | C32⋊C4 | C2×C32⋊C4 | C3⋊S3⋊3C8 | S3×C32⋊C4 | C33⋊5(C2×C8) |
| kernel | C33⋊5(C2×C8) | C3×C32⋊2C8 | C33⋊4C8 | C33⋊8(C2×C4) | C32×Dic3 | C2×C33⋊C2 | C33⋊C2 | C32⋊2C8 | C3⋊Dic3 | C3×C6 | C32 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 2 |
Matrix representation of C33⋊5(C2×C8) ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 72 | 71 | 72 | 72 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 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 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 72 |
| 0 | 0 | 0 | 72 | 72 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 10 | 0 |
| 0 | 0 | 0 | 0 | 63 | 10 |
| 0 | 0 | 0 | 0 | 0 | 63 |
| 0 | 0 | 0 | 63 | 0 | 63 |
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] >;
C33⋊5(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