Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#5931 |
| Fitting | C23×Q8 ⋊ D3 |
| derived | C22⋊SL(2,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #1020 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C22 |
| Frattini | C2 | C2×#1020 |
| Fitting | C23×Q8 | D3 |
| socle | C24 | S4 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
310 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 48 |
A minimal faithful permutation representation — 2 generators on 14 points:
(3,4,6,5)(7,8,13,9,10,12,11,14);(1,2)(4,5,6)(8,11,14)(9,12,13)30 conjugacy classes — view the full character table.
| maximal subgroup of | #11711140, #11711141, #11711237, … |
| maximal normal subgroup of | #11711140, #11711141, #11711237, … |
| maximal quotient of | C2×#85551, C2×#85548, #11715579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20082);