Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C2×S4)⋊C8 |
| Fitting | C23×C8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×S4 |
| commutator | C2×A4 | C2×C8 |
| Fitting | C23×C8 | D3 |
| socle | C24 | C4×D3 |
| prime radical | C22×S4 | C4 |
| cosocle | C2×C4×A4 | C22 |
| hypercenter | C2×C8 | S4 |
| hypocenter | A4 | C22⋊C8 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8 | |
217 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,4)(5,6,8)(9,10,11,13,12,14,15,16);(1,2)(3,4)(6,7)Polycyclic presentation:
2^5,3,2^2|b^2=c,c^2=e,b^a=bd,f^a=f^2,g^a=h,h^a=g,g^f=h,h^f=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×(C2×S4)⋊C8, #11712705, #11712706, … |
| maximal normal subgroup of | C2×(C2×S4)⋊C8, #11712705, #11712706, … |
| maximal quotient of | #11711104, #11711105, #11711106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5688);