Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#387 ⋊ C2 |
| Sylow | C3 ⋊ #2368 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#387 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C23×D3 | C23 |
| hypercenter | #387 | D3 |
| hypocenter | C3 | #2368 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #387 | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(10,15)(12,14);(2,3)(9,13)(12,14)(16,18,19,23)(17,20,21,22);(4,6,7,5)(8,10,11,15)(9,12,13,14);(2,3)(4,5,7,6)(8,9)(10,14)(11,13)(12,15)(16,17,19,21)(18,22,23,20)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=efg,b^a=be,c^a=cfg,h^a=h^2,d^b=de,h^c=h^2,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2368, C2×#75311, #10954846, … |
| maximal normal subgroup of | D3×#2368, C2×#75311, #10954846, … |
| maximal quotient of | #10716617, #10716692, #10716699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7887);