Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4626 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C2×#4626 |
| derived | C42⋊C3 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4626 |
| commutator, hypocenter | C42⋊C3 | C2×C4 |
| Frattini | C23 | C2×S4 |
| Fitting | C2×#127 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical | C2×#127⋊C3 | C2 |
| cosocle | #127⋊C3 | C22 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 | |
130 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 18 points:
(3,12,17,16)(4,18,9,10)(5,6,13,15)(7,8,11,14);(1,2)(4,9,11)(5,15,12)(6,7,10)(8,16,17)(13,14,18)22 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4626, #11711503, (#560⋊C3)⋊C4, … |
| maximal normal subgroup of | C4×#4626, #11711503, (#560⋊C3)⋊C4, … |
| maximal quotient of | C2×#67999, #11711341, #11711343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5658);