Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #120⋊C3 |
| automorphism | Aut(G) = C2×#4625⋊C2 (order 768) · Inn(G) = #120⋊C3 · Out(G) = C23 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #379⋊C3 |
| Fitting, Sylow | #379 ⋊ C3 |
| derived | C24 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #120⋊C3 |
| commutator, hypocenter | C24 | C3×C4 |
| Frattini, socle | C23 | C2×A4 |
| Fitting | #379 | C3 |
| prime radical | C2×C22⋊A4 | C2 |
| cosocle | C25 | C2×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,5,11,6,3,9,4,10,7,2,8,12);(2,6)(9,12)Polycyclic presentation:
2,3,2^5|a^2=c,d^a=df,e^a=eg,d^b=e,e^b=de,f^b=g,g^b=fg20 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 191);