Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2329 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#380 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | #380 | D3 |
| hypocenter | C3 | #2329 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #380 | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,13)(12,14);(10,15)(12,14)(16,18,19,23)(17,20,21,22);(2,3)(4,6,7,5)(8,10)(9,12)(11,15)(13,14);(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=fg,b^a=be,c^a=cfg,d^a=de,c^b=ce,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10689239, #10689243, #10689246, … |
| maximal normal subgroup of | #10689239, #10689243, #10689246, … |
| maximal quotient of | #10716442, #10716469, #10716561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6450);