Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C23×#237 |
| Fitting | C23×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ C23×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C2×C3 | C26 |
| Frattini | C2 | C25×D3 |
| Fitting | C23×C3×D4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | C23×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
6520 conjugacy classes of subgroups; only the 3301 normal subgroups are shown.
This group has 6520 conjugacy classes of subgroups — too many to draw. Only its 3301 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 17 points:
(1,2)(3,4)(5,6)(7,8,9)(11,15)(12,16);(10,12,13,16)(11,14,15,17);(8,9)(10,11)(12,14)(13,15)(16,17);(5,6);(3,4);(1,2)120 conjugacy classes — view the full character table.
| maximal subgroup of | #10716945, #10717702, #10717705, … |
| maximal normal subgroup of | #10716945, #10717702, #10717705, … |
| maximal quotient of | C23×#877, C23×#892, C23×#67⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20132);