metabelian, supersoluble, monomial
Aliases: C3⋊C8⋊20D6, C12.32(C4×S3), C3⋊S3⋊2M4(2), C3⋊3(S3×M4(2)), C4.Dic3⋊6S3, (C2×C12).109D6, C62.48(C2×C4), C32⋊6(C2×M4(2)), (C6×C12).69C22, C4.9(C6.D6), C12.31D6⋊12C2, C12.29D6⋊11C2, 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
Generators and relations for C3⋊C8⋊20D6
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, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C2×M4(2), C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C3×C3⋊C8, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×M4(2), C12.29D6, C12.31D6, C3×C4.Dic3, C2×C4×C3⋊S3, C3⋊C8⋊20D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4×S3, C22×S3, C2×M4(2), S32, S3×C2×C4, C6.D6, C2×S32, S3×M4(2), C2×C6.D6, C3⋊C8⋊20D6
►On 24 points - transitive group 24T616
(1 21 15)(2 16 22)(3 23 9)(4 10 24)(5 17 11)(6 12 18)(7 19 13)(8 14 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 15 21)(2 12 22 6 16 18)(3 9 23)(4 14 24 8 10 20)(5 11 17)(7 13 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)| (1,21,15)(2,16,22)(3,23,9)(4,10,24)(5,17,11)(6,12,18)(7,19,13)(8,14,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,15,21)(2,12,22,6,16,18)(3,9,23)(4,14,24,8,10,20)(5,11,17)(7,13,19), (1,21)(2,18)(3,23)(4,20)(5,17)(6,22)(7,19)(8,24)(10,14)(12,16)>;
G:=Group( (1,21,15)(2,16,22)(3,23,9)(4,10,24)(5,17,11)(6,12,18)(7,19,13)(8,14,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,15,21)(2,12,22,6,16,18)(3,9,23)(4,14,24,8,10,20)(5,11,17)(7,13,19), (1,21)(2,18)(3,23)(4,20)(5,17)(6,22)(7,19)(8,24)(10,14)(12,16) );
G=PermutationGroup([[(1,21,15),(2,16,22),(3,23,9),(4,10,24),(5,17,11),(6,12,18),(7,19,13),(8,14,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,15,21),(2,12,22,6,16,18),(3,9,23),(4,14,24,8,10,20),(5,11,17),(7,13,19)], [(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 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | ··· | 6G | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 9 | 9 | 18 | 2 | 2 | 4 | 1 | 1 | 2 | 9 | 9 | 18 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | S32 | C6.D6 | C2×S32 | C6.D6 | S3×M4(2) | C3⋊C8⋊20D6 |
| kernel | C3⋊C8⋊20D6 | C12.29D6 | C12.31D6 | C3×C4.Dic3 | C2×C4×C3⋊S3 | C4×C3⋊S3 | C2×C3⋊Dic3 | C22×C3⋊S3 | C4.Dic3 | C3⋊C8 | C2×C12 | C3⋊S3 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 |
Matrix representation of C3⋊C8⋊20D6 ►in GL4(𝔽5) generated by
| 4 | 0 | 3 | 0 |
| 0 | 4 | 0 | 2 |
| 3 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 2 |
| 2 | 0 | 4 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 4 | 0 | 2 |
| 3 | 0 | 1 | 0 |
| 0 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 2 | 0 | 4 | 0 |
| 0 | 2 | 0 | 1 |
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⋊C8⋊20D6 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