Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #127⋊C3 |
| automorphism | Aut(G) = C2×#4626⋊C2 (order 768) · Inn(G) = #127⋊C3 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
| direct product | C2×#127⋊C3 |
| unambiguous semidirect product | (C2×#127)⋊C3 |
| Fitting, Sylow | C2×#127 ⋊ C3 |
| derived | C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #127⋊C3 |
| commutator, cosocle, hypocenter | C42 | C22×C3 |
| Frattini | C22 | C22×A4 |
| Fitting | C2×#127 | C3 |
| socle | C23 | C2×A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C2 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,3,7)(2,5,9)(4,10,11,6,8,12);(1,2)(3,10,5,8)(4,6)(7,11,9,12)20 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4626, #73083, #73084, … |
| maximal normal subgroup of | C2×#4626, #73083, #73084, … |
| maximal quotient of | C2×#384⋊C3, C4×#127⋊C3, C2×#392⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1002);