Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #1020⋊C2 |
| automorphism | Aut(G) = #4625⋊C2 (order 384) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #1020⋊C2 |
| Fitting | #120 ⋊ D3 |
| derived | C22⋊A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊A4 | C22 |
| Frattini, socle | C22 | C2×S4 |
| Fitting | #120 | D3 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 | |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,5,3,6,4,2)(7,8);(3,8)(5,6)Polycyclic presentation:
2^2,3,2^4|c^a=c^2,d^a=e,e^a=d,f^a=fg,d^b=dfg,e^b=ef,d^c=e,e^c=de,f^c=g,g^c=fg14 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 955);