Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #204 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#904 |
| Fitting | C23×C3×C8 ⋊ C2 |
| Sylow | C3 ⋊ C22×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D12 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C23×C3×C8 | C2 |
| socle | C23×C3 | C2×D4 |
| hypercenter | C23×C8 | D3 |
| hypocenter | C3 | C22×#135 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C23×C8 | |
1504 conjugacy classes of subgroups; only the 565 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 565 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(5,6,7)(8,11,12,19,13,20,10,17)(9,15,14,18,16,22,21,23);(1,2)(3,4)(8,10,13,12)(9,14,16,21)(11,17,20,19)(15,18,22,23);(6,7)(8,9)(10,14)(11,18)(12,21)(13,16)(15,19)(17,22)(20,23);(3,4);(1,2)120 conjugacy classes — view the full character table.
| maximal subgroup of | #10702002, #10702004, #10702006, … |
| maximal normal subgroup of | #10702002, #10702004, #10702006, … |
| maximal quotient of | C22×#5113, C22×#5106, C22×C4×D24, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19726);