direct product, metabelian, soluble, monomial, A-group
Aliases: C22×C42⋊C3, C24.9A4, (C2×C42)⋊3C6, C42⋊16(C2×C6), (C22×C42)⋊1C3, C23.20(C2×A4), C22.1(C22×A4), SmallGroup(192,992)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C22×C42⋊C3 |
Generators and relations for C22×C42⋊C3
G = < a,b,c,d,e | a2=b2=c4=d4=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, ede-1=c-1d2 >
Subgroups: 354 in 108 conjugacy classes, 20 normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C2×C4, C23, C23, A4, C2×C6, C42, C42, C22×C4, C24, C2×A4, C2×C42, C2×C42, C23×C4, C42⋊C3, C22×A4, C22×C42, C2×C42⋊C3, C22×C42⋊C3
Quotients: C1, C2, C3, C22, C6, A4, C2×C6, C2×A4, C42⋊C3, C22×A4, C2×C42⋊C3, C22×C42⋊C3
►On 24 points - transitive group 24T420
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 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)
(1 6 8 3)(2 5 7 4)(9 14 11 16)(10 15 12 13)(17 23)(18 24)(19 21)(20 22)
(1 24 9)(2 20 13)(3 17 12)(4 21 16)(5 23 14)(6 19 10)(7 18 15)(8 22 11)
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,8)(2,7)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,8,3)(2,5,7,4)(9,14,11,16)(10,15,12,13)(17,23)(18,24)(19,21)(20,22), (1,24,9)(2,20,13)(3,17,12)(4,21,16)(5,23,14)(6,19,10)(7,18,15)(8,22,11)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24), (1,8)(2,7)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,8,3)(2,5,7,4)(9,14,11,16)(10,15,12,13)(17,23)(18,24)(19,21)(20,22), (1,24,9)(2,20,13)(3,17,12)(4,21,16)(5,23,14)(6,19,10)(7,18,15)(8,22,11) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24)], [(1,8),(2,7),(3,6),(4,5),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,6,8,3),(2,5,7,4),(9,14,11,16),(10,15,12,13),(17,23),(18,24),(19,21),(20,22)], [(1,24,9),(2,20,13),(3,17,12),(4,21,16),(5,23,14),(6,19,10),(7,18,15),(8,22,11)]])
G:=TransitiveGroup(24,420);
►On 24 points - transitive group 24T421
Generators in S24
(1 7)(2 8)(3 5)(4 6)(9 22)(10 23)(11 24)(12 21)(13 19)(14 20)(15 17)(16 18)
(1 8)(2 7)(3 6)(4 5)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(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 6 2 5)(3 7 4 8)(13 16 15 14)(17 20 19 18)
(1 24 17)(2 22 19)(3 10 14)(4 12 16)(5 23 20)(6 21 18)(7 11 15)(8 9 13)
G:=sub<Sym(24)| (1,7)(2,8)(3,5)(4,6)(9,22)(10,23)(11,24)(12,21)(13,19)(14,20)(15,17)(16,18), (1,8)(2,7)(3,6)(4,5)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(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), (1,6,2,5)(3,7,4,8)(13,16,15,14)(17,20,19,18), (1,24,17)(2,22,19)(3,10,14)(4,12,16)(5,23,20)(6,21,18)(7,11,15)(8,9,13)>;
G:=Group( (1,7)(2,8)(3,5)(4,6)(9,22)(10,23)(11,24)(12,21)(13,19)(14,20)(15,17)(16,18), (1,8)(2,7)(3,6)(4,5)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(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), (1,6,2,5)(3,7,4,8)(13,16,15,14)(17,20,19,18), (1,24,17)(2,22,19)(3,10,14)(4,12,16)(5,23,20)(6,21,18)(7,11,15)(8,9,13) );
G=PermutationGroup([[(1,7),(2,8),(3,5),(4,6),(9,22),(10,23),(11,24),(12,21),(13,19),(14,20),(15,17),(16,18)], [(1,8),(2,7),(3,6),(4,5),(9,24),(10,21),(11,22),(12,23),(13,17),(14,18),(15,19),(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)], [(1,6,2,5),(3,7,4,8),(13,16,15,14),(17,20,19,18)], [(1,24,17),(2,22,19),(3,10,14),(4,12,16),(5,23,20),(6,21,18),(7,11,15),(8,9,13)]])
G:=TransitiveGroup(24,421);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4P | 6A | ··· | 6F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 16 | 16 | 3 | ··· | 3 | 16 | ··· | 16 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C3 | C6 | A4 | C2×A4 | C42⋊C3 | C2×C42⋊C3 |
| kernel | C22×C42⋊C3 | C2×C42⋊C3 | C22×C42 | C2×C42 | C24 | C23 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 1 | 3 | 4 | 12 |
Matrix representation of C22×C42⋊C3 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,5,0,0,0,0,8,0,0,0,0,1],[3,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
C22×C42⋊C3 in GAP, Magma, Sage, TeX
C_2^2\times C_4^2\rtimes C_3
% in TeX
G:=Group("C2^2xC4^2:C3"); // GroupNames label
G:=SmallGroup(192,992);
// by ID
G=gap.SmallGroup(192,992);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,185,360,1173,102,1027,1784]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^4=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,e*d*e^-1=c^-1*d^2>;
// generators/relations