Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 147456 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×D3×D4 |
| Fitting | C22×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ C23×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D3 |
| commutator, cosocle | C2×C3 | C25 |
| Frattini | C2 | C24×D3 |
| Fitting | C22×C3×D4 | C2 |
| socle | C23×C3 | C23 |
| hypercenter | C22×D4 | D3 |
| hypocenter | C3 | C23×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
1362 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 503 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 5 generators on 11 points:
(1,2)(3,4)(5,6,7)(8,9)(10,11);(1,2)(3,4)(9,11);(6,7);(3,4);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | #70035, #70079, #70198, … |
| maximal normal subgroup of | #70035, #70079, #70198, … |
| maximal quotient of | C22×#67⋊C4, C22×#878, C22×#885, … |
| automorphism group of | #805, #822, #825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1514);