Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×C8×D4 ⋊ C2 |
| Sylow | C3 ⋊ #2933 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×C8×D4 | C2 |
| prime radical | #5547 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | C8×D4 | D3 |
| hypocenter | C3 | #2933 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C8×D4 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,3)(10,16)(13,19)(17,22)(20,23);(1,2,3)(8,10,11,17,12,16,18,22)(9,13,14,20,15,19,21,23);(5,7);(4,5)(6,7)(8,9,11,14,12,15,18,21)(10,13,17,20,16,19,22,23)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,f^2=g,b^a=be,d^c=dg,h^d=h^290 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2933, C2×#79423, #11659110, … |
| maximal normal subgroup of | D3×#2933, C2×#79423, #11659110, … |
| maximal quotient of | #10766068, #10766079, #10766090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11999);