direct product, metabelian, soluble, monomial
Aliases: A4×C4○D4, D4⋊2(C2×A4), (D4×A4)⋊5C2, (Q8×A4)⋊5C2, Q8⋊4(C2×A4), C24.(C2×C6), (C23×C4)⋊3C6, (C22×D4)⋊3C6, C2.6(C23×A4), C4.8(C22×A4), (C22×Q8)⋊7C6, (C4×A4).21C22, (C2×A4).14C23, C22.7(C22×A4), (C22×A4).2C22, C23.31(C22×C6), (C2×C4×A4)⋊7C2, (C2×C4)⋊3(C2×A4), C22⋊(C3×C4○D4), (C22×C4○D4)⋊2C3, (C22×C4).5(C2×C6), SmallGroup(192,1501)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 620 in 215 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2 [×8], C3, C4, C4 [×3], C4 [×4], C22, C22 [×3], C22 [×17], C6 [×4], C2×C4 [×3], C2×C4 [×23], D4 [×3], D4 [×15], Q8, Q8 [×5], C23, C23 [×12], C12 [×4], A4, C2×C6 [×3], C22×C4, C22×C4 [×3], C22×C4 [×12], C2×D4 [×12], C2×Q8 [×4], C4○D4, C4○D4 [×21], C24 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C2×A4, C2×A4 [×3], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×8], C4×A4, C4×A4 [×3], C3×C4○D4, C22×A4 [×3], C22×C4○D4, C2×C4×A4 [×3], D4×A4 [×3], Q8×A4, A4×C4○D4
Quotients:
C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, A4, C2×C6 [×7], C4○D4, C2×A4 [×7], C22×C6, C3×C4○D4, C22×A4 [×7], C23×A4, A4×C4○D4
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c3=d4=f2=1, e2=d2, 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, de=ed, df=fd, fef=d2e >
►On 24 points - transitive group 24T296
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(1 13 11)(2 14 12)(3 15 9)(4 16 10)(5 18 21)(6 19 22)(7 20 23)(8 17 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 23 3 21)(2 24 4 22)(5 13 7 15)(6 14 8 16)(9 18 11 20)(10 19 12 17)
(5 7)(6 8)(17 19)(18 20)(21 23)(22 24)
G:=sub<Sym(24)| (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (1,13,11)(2,14,12)(3,15,9)(4,16,10)(5,18,21)(6,19,22)(7,20,23)(8,17,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,23,3,21)(2,24,4,22)(5,13,7,15)(6,14,8,16)(9,18,11,20)(10,19,12,17), (5,7)(6,8)(17,19)(18,20)(21,23)(22,24)>;
G:=Group( (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (1,13,11)(2,14,12)(3,15,9)(4,16,10)(5,18,21)(6,19,22)(7,20,23)(8,17,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,23,3,21)(2,24,4,22)(5,13,7,15)(6,14,8,16)(9,18,11,20)(10,19,12,17), (5,7)(6,8)(17,19)(18,20)(21,23)(22,24) );
G=PermutationGroup([(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(9,11),(10,12),(17,19),(18,20),(21,23),(22,24)], [(1,13,11),(2,14,12),(3,15,9),(4,16,10),(5,18,21),(6,19,22),(7,20,23),(8,17,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,23,3,21),(2,24,4,22),(5,13,7,15),(6,14,8,16),(9,18,11,20),(10,19,12,17)], [(5,7),(6,8),(17,19),(18,20),(21,23),(22,24)])
G:=TransitiveGroup(24,296);
Matrix representation ►G ⊆ GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 12 | 12 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 4 | 4 | 4 |
| 0 | 0 | 0 | 9 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 5 | 9 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 8 | 4 | 0 | 0 | 0 |
| 7 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,1,0,12,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,12,0,0,0,1,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,4,0,0,0,0,4,9,0,0,0,4,0],[8,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[5,0,0,0,0,9,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,7,0,0,0,4,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | ··· | 6H | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 4 | 4 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 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 | C2 | C3 | C6 | C6 | C6 | C4○D4 | C3×C4○D4 | A4 | C2×A4 | C2×A4 | C2×A4 | A4×C4○D4 |
| kernel | A4×C4○D4 | C2×C4×A4 | D4×A4 | Q8×A4 | C22×C4○D4 | C23×C4 | C22×D4 | C22×Q8 | A4 | C22 | C4○D4 | C2×C4 | D4 | Q8 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 2 | 4 | 1 | 3 | 3 | 1 | 2 |
In GAP, Magma, Sage, TeX
A_4\times C_4\circ D_4
% in TeX
G:=Group("A4xC4oD4"); // GroupNames label
G:=SmallGroup(192,1501);
// by ID
G=gap.SmallGroup(192,1501);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,2,176,590,530,909]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^4=f^2=1,e^2=d^2,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,d*e=e*d,d*f=f*d,f*e*f=d^2*e>;
// generators/relations