Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#5936 |
| derived | C22⋊A4 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #1020 |
| commutator, hypocenter | C22⋊A4 | C2×C4 |
| Frattini | C2 | C2×#1020 |
| Fitting, socle | C26 | D3 |
| prime radical | C22×C22⋊A4 | C2 |
| cosocle | C2×C22⋊A4 | C22 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C22 | |
847 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 847 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 2 generators on 14 points:
(3,4,6,5)(7,8,10,9)(11,12,13,14);(1,2)(4,6,5)(8,9,10)40 conjugacy classes — view the full character table.
| maximal subgroup of | #11711329, #11711331, #11711332, … |
| maximal normal subgroup of | #11711329, #11711331, #11711332, … |
| maximal quotient of | C4×#5936, #11715569, C4⋊#5936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20099);