Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #857 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×#4623 |
| derived | C42⋊C3 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #857 |
| commutator, hypocenter | C42⋊C3 | C2×C4 |
| Frattini | C23 | C2×S4 |
| Fitting | C22×C42 | D3 |
| socle | C24 | S4 |
| prime radical | C22×C42⋊C3 | C2 |
| cosocle | C2×C42⋊C3 | C22 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C22 | |
190 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 190 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) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(3,5)(4,7)(6,11,8,10,9,13,12,14);(1,2)(3,6,11)(4,8,10)(5,9,13)(7,12,14)40 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4623, #11711290, C4⋊#4623, … |
| maximal normal subgroup of | C4×#4623, #11711290, C4⋊#4623, … |
| maximal quotient of | C2×#67995, #11710976, C2×SL(2,ℤ/8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5623);