Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C22×#1004 |
| Fitting | C23×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ C22×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C2×C3 | C26 |
| Frattini | C2 | C25×D3 |
| Fitting | C23×C3×D4 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | C22×H(5,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.
This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 15 points:
(1,2)(3,4)(5,6,7)(8,10)(9,13)(11,14)(12,15);(8,9)(10,13)(11,12)(14,15);(9,12)(13,15);(6,7)(10,14)(13,15);(3,4);(1,2)108 conjugacy classes — view the full character table.
| maximal subgroup of | #10794320, #10794321, #10794323, … |
| maximal normal subgroup of | #10794320, #10794321, #10794323, … |
| maximal quotient of | C22×#5545, C22×#5486, C22×#5798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20134);