Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#1020 |
| Fitting | C24×C4 ⋊ D3 |
| derived | C22⋊A4 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #1020 |
| commutator, hypocenter | C22⋊A4 | C2×C4 |
| Frattini | C2 | C2×#1020 |
| Fitting | C24×C4 | D3 |
| socle | C25 | C2×D3 |
| prime radical | C2×#1020 | C2 |
| cosocle | C2×C22⋊A4 | C22 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C4 | |
435 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(5,6,8,7)(9,10,12,11);(1,2,3,4)(6,7,8)(10,12,11)40 conjugacy classes — view the full character table.
| maximal subgroup of | #1020⋊C8, #11711114, C4×#1020⋊C2, … |
| maximal normal subgroup of | #1020⋊C8, #11711114, C4×#1020⋊C2, … |
| maximal quotient of | C4×#5936, #11715571, #11715572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20079);