Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #857 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C22×#857 |
| Fitting | C22×C42 ⋊ D3 |
| derived | C42⋊C3 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #857 |
| commutator, cosocle, hypocenter | C42⋊C3 | C23 |
| Frattini | C22 | C22×S4 |
| Fitting | C22×C42 | D3 |
| socle | C24 | S4 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C22 | |
398 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 24 |
A minimal faithful permutation representation — 3 generators on 14 points:
(5,7)(6,11,12,10,8,14,9,13);(3,8,11,4,6,14)(5,12,10,7,9,13);(1,2)40 conjugacy classes — view the full character table.
| maximal subgroup of | #11711328, #11711335, C2×C4×#857, … |
| maximal normal subgroup of | #11711328, #11711335, C2×C4×#857, … |
| maximal quotient of | C22×#4622, C22×#4621, C2×#72987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17861);