Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C26 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#1020 |
| commutator, cosocle | C2×C22⋊A4 | C22 |
| Fitting | C26 | D3 |
| socle | C25 | C2×D3 |
| hypercenter | C22 | #1020 |
| hypocenter | C22⋊A4 | D4 |
| LCS | G ▷ C2×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 ◁ C22 | |
826 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 826 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,7,9)(2,6,11,5,10,8)(3,4,12);(1,5,3,2)(4,11)(6,12,10,9)(7,8)Polycyclic presentation:
2^3,3,2^4|b^a=bc,d^a=d^2,f^a=ef,g^a=gh,e^d=f,f^d=ef,g^d=h,h^d=gh28 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20100);