Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) order 73728 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C42×D3 |
| Fitting | C2×C3×C42 ⋊ C2 |
| derived, Sylow | C3 ⋊ C22×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | D3 |
| commutator, hypocenter | C3 | C22×C42 |
| Frattini | C22 | C23×D3 |
| Fitting | C2×C3×C42 | C2 |
| socle | C23×C3 | C23 |
| prime radical | C23×D3 | C22 |
| cosocle | C22×C3 | C24 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C42 | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2)(4,5);(3,4,5)(6,7,8,9);(10,11,12,13);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #68599, #68600, D3×#336, … |
| maximal normal subgroup of | #68599, #68600, D3×#336, … |
| maximal quotient of | C4×#872, C42×D12, C42×Dic6, … |
| automorphism group of | C5×C3⋊C16 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1030);