Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#436 ⋊ C2 |
| Sylow | C3 ⋊ #2933 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#436 | C2 |
| prime radical | #5547 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #436 | D3 |
| hypocenter | C3 | #2933 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #436 | |
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:
(1,2)(10,16)(13,19)(17,22)(20,23);(8,10,11,17,12,16,18,22)(9,13,14,20,15,19,21,23);(2,3)(5,7);(2,3)(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,h^a=h^2,h^b=h^2,d^c=dg,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2933, C2×#79539, #11659111, … |
| maximal normal subgroup of | D3×#2933, C2×#79539, #11659111, … |
| maximal quotient of | #10766077, #10766088, #10766102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12115);