Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C4⋊(A4⋊C8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×S4 |
| commutator | C2×A4 | C2×C8 |
| Fitting | C22×C42 | D3 |
| socle | C24 | C2×Dic3 |
| prime radical | C22×A4 | C2×C4 |
| cosocle | C2×C4×A4 | C22 |
| hypercenter | C42 | S4 |
| hypocenter | A4 | C4⋊C8 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
165 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 96 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4)(5,6,8);(2,4)(6,7)(9,10,11,13,12,14,15,16)Polycyclic presentation:
2^5,3,2^2|a^2=c,b^2=d,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 | D4⋊(A4⋊C8), #11711012, #11711023, … |
| maximal normal subgroup of | D4⋊(A4⋊C8), #11711012, #11711023, … |
| maximal quotient of | C4⋊(SL(2,𝔽3)⋊C8), #11711065, #11711066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5621);