Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #5386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#5421 |
| Fitting | C23×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Frattini | C22 | C22×S4 |
| Fitting | C23×Q8 | D3 |
| hypercenter | C24 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C24 | |
427 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 14 points:
(3,4)(5,6)(7,8,12)(10,13,14);(4,6)(8,9)(11,13)(12,14);(1,2)46 conjugacy classes — view the full character table.
| maximal subgroup of | #11711199, C4×#5421, #11712808, … |
| maximal normal subgroup of | #11711199, C4×#5421, #11712808, … |
| maximal quotient of | C2×#73088, C2×#73087, C2×#73100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18081);