Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #126⋊C3 |
| automorphism | Aut(G) = C2×#560⋊C3 (order 384) · Inn(G) = #126⋊C3 · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×#126⋊C3 |
| unambiguous semidirect product | (C2×#126)⋊C3 |
| Fitting, Sylow | C2×#126 ⋊ C3 |
| derived | C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #126⋊C3 |
| commutator, cosocle, hypocenter | C42 | C22×C3 |
| Frattini | C22 | C22×A4 |
| Fitting | C2×#126 | C3 |
| socle | C23 | C2×A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C2 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(4,12,13,8,5,15)(6,7,9)(10,11,14,18,16,17);(1,2)(3,15,7,18)(4,12,11,5)(6,10,9,13)(8,17,16,14)20 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#126⋊C3, (#126⋊C3)⋊C4, #2818⋊C3, … |
| maximal normal subgroup of | C4×#126⋊C3, (#126⋊C3)⋊C4, #2818⋊C3, … |
| maximal quotient of | C2×#383⋊C3, C4×#126⋊C3, C2×#396⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1001);