Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = S4×#120 (order 768) · Inn(G) = C2×S4 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C4⋊SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×S4 |
| commutator | C2×A4 | C2×C4 |
| Fitting | C23×C4 | D3 |
| socle | C24 | C2×D3 |
| prime radical, cosocle | C22×A4 | C22 |
| hypercenter | C2×C4 | S4 |
| hypocenter | A4 | C4⋊C4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 12 points:
(5,6,7,8)(9,10,12);(1,2,3,4)(6,8)(10,11)Polycyclic presentation:
2^4,3,2^2|a^2=c,b^2=d,b^a=bd,e^a=e^2,f^a=g,g^a=f,f^e=g,g^e=fg28 conjugacy classes — view the full character table.
| maximal subgroup of | #73050, #73051, #73109, … |
| maximal normal subgroup of | #73050, #73051, #73109, … |
| maximal quotient of | C4⋊(A4⋊C8), C4⋊#859, #73064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 971);