direct product, metabelian, soluble, monomial
Aliases: A4×C22⋊C4, C24⋊2C12, C25.2C6, C2.1(D4×A4), C22⋊2(C4×A4), (C23×C4)⋊1C6, (C22×A4)⋊1C4, (C2×A4).12D4, (C23×A4).1C2, C24.24(C2×C6), C23.24(C2×A4), C23.21(C3×D4), C23.15(C2×C12), C22.11(C22×A4), (C22×A4).14C22, (C2×C4×A4)⋊2C2, C2.3(C2×C4×A4), (C2×C4)⋊1(C2×A4), C22⋊(C3×C22⋊C4), (C22×C22⋊C4)⋊C3, (C2×A4).11(C2×C4), SmallGroup(192,994)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for A4×C22⋊C4
G = < a,b,c,d,e,f | a2=b2=c3=d2=e2=f4=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, ef=fe >
Subgroups: 620 in 169 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C2×C4, C2×C4, C23, C23, C23, C12, A4, C2×C6, C22⋊C4, C22⋊C4, C22×C4, C24, C24, C24, C2×C12, C2×A4, C2×A4, C2×A4, C22×C6, C2×C22⋊C4, C23×C4, C25, C3×C22⋊C4, C4×A4, C22×A4, C22×A4, C22×A4, C22×C22⋊C4, C2×C4×A4, C23×A4, A4×C22⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, A4, C2×C6, C22⋊C4, C2×C12, C3×D4, C2×A4, C3×C22⋊C4, C4×A4, C22×A4, C2×C4×A4, D4×A4, A4×C22⋊C4
►On 24 points - transitive group 24T408
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 7)(2 8)(3 5)(4 6)(9 15)(10 16)(11 13)(12 14)(17 19)(18 20)(21 23)(22 24)
(1 15 23)(2 16 24)(3 13 21)(4 14 22)(5 9 17)(6 10 18)(7 11 19)(8 12 20)
(1 3)(2 8)(4 6)(5 7)(9 11)(10 14)(12 16)(13 15)(17 19)(18 22)(20 24)(21 23)
(1 5)(2 6)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)
(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)| (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,7)(2,8)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14)(17,19)(18,20)(21,23)(22,24), (1,15,23)(2,16,24)(3,13,21)(4,14,22)(5,9,17)(6,10,18)(7,11,19)(8,12,20), (1,3)(2,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (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( (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,7)(2,8)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14)(17,19)(18,20)(21,23)(22,24), (1,15,23)(2,16,24)(3,13,21)(4,14,22)(5,9,17)(6,10,18)(7,11,19)(8,12,20), (1,3)(2,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (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([[(1,5),(2,6),(3,7),(4,8),(9,11),(10,12),(13,15),(14,16),(17,21),(18,22),(19,23),(20,24)], [(1,7),(2,8),(3,5),(4,6),(9,15),(10,16),(11,13),(12,14),(17,19),(18,20),(21,23),(22,24)], [(1,15,23),(2,16,24),(3,13,21),(4,14,22),(5,9,17),(6,10,18),(7,11,19),(8,12,20)], [(1,3),(2,8),(4,6),(5,7),(9,11),(10,14),(12,16),(13,15),(17,19),(18,22),(20,24),(21,23)], [(1,5),(2,6),(3,7),(4,8),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(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(24,408);
►On 24 points - transitive group 24T409
Generators in S24
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(1 19 6)(2 20 7)(3 17 8)(4 18 5)(9 15 21)(10 16 22)(11 13 23)(12 14 24)
(1 3)(2 10)(4 12)(5 24)(6 8)(7 22)(9 11)(13 15)(14 18)(16 20)(17 19)(21 23)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 19)(14 20)(15 17)(16 18)
(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)| (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,15,21)(10,16,22)(11,13,23)(12,14,24), (1,3)(2,10)(4,12)(5,24)(6,8)(7,22)(9,11)(13,15)(14,18)(16,20)(17,19)(21,23), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18), (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( (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,15,21)(10,16,22)(11,13,23)(12,14,24), (1,3)(2,10)(4,12)(5,24)(6,8)(7,22)(9,11)(13,15)(14,18)(16,20)(17,19)(21,23), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18), (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([[(5,7),(6,8),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(21,23),(22,24)], [(1,19,6),(2,20,7),(3,17,8),(4,18,5),(9,15,21),(10,16,22),(11,13,23),(12,14,24)], [(1,3),(2,10),(4,12),(5,24),(6,8),(7,22),(9,11),(13,15),(14,18),(16,20),(17,19),(21,23)], [(1,11),(2,12),(3,9),(4,10),(5,22),(6,23),(7,24),(8,21),(13,19),(14,20),(15,17),(16,18)], [(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(24,409);
►On 24 points - transitive group 24T410
Generators in S24
(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 12)(3 9)(4 10)(17 21)(18 22)(19 23)(20 24)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 24)
(2 12)(4 10)(6 13)(8 15)(18 22)(20 24)
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 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)| (5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(17,21)(18,22)(19,23)(20,24), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24), (2,12)(4,10)(6,13)(8,15)(18,22)(20,24), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (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( (5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(17,21)(18,22)(19,23)(20,24), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24), (2,12)(4,10)(6,13)(8,15)(18,22)(20,24), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (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([[(5,16),(6,13),(7,14),(8,15),(17,21),(18,22),(19,23),(20,24)], [(1,11),(2,12),(3,9),(4,10),(17,21),(18,22),(19,23),(20,24)], [(1,7,19),(2,8,20),(3,5,17),(4,6,18),(9,16,21),(10,13,22),(11,14,23),(12,15,24)], [(2,12),(4,10),(6,13),(8,15),(18,22),(20,24)], [(1,11),(2,12),(3,9),(4,10),(5,16),(6,13),(7,14),(8,15),(17,21),(18,22),(19,23),(20,24)], [(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(24,410);
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | C3×D4 | A4 | C2×A4 | C2×A4 | C4×A4 | D4×A4 |
| kernel | A4×C22⋊C4 | C2×C4×A4 | C23×A4 | C22×C22⋊C4 | C22×A4 | C23×C4 | C25 | C24 | C2×A4 | C23 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 4 | 1 | 2 | 1 | 4 | 2 |
Matrix representation of A4×C22⋊C4 ►in GL7(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12] >;
A4×C22⋊C4 in GAP, Magma, Sage, TeX
A_4\times C_2^2\rtimes C_4
% in TeX
G:=Group("A4xC2^2:C4"); // GroupNames label
G:=SmallGroup(192,994);
// by ID
G=gap.SmallGroup(192,994);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,365,92,1027,1784]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^2=e^2=f^4=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations