Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#399 ⋊ C2 |
| Sylow | C3 ⋊ #2718 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#399 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #399 | D3 |
| hypocenter | C3 | #2718 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #399 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,3)(10,15)(12,14)(17,22)(21,23);(1,2,3)(5,7)(9,13)(10,15)(16,19,20,18)(17,21,22,23);(4,5)(6,7)(8,10)(9,12)(11,15)(13,14)(16,18,20,19)(17,21,22,23);(8,9,11,13)(10,14,15,12)(16,17)(18,21)(19,23)(20,22)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,c^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #11002586, #11002592, #11002662, … |
| maximal normal subgroup of | #11002586, #11002592, #11002662, … |
| maximal quotient of | #10722840, #10722849, #10722909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10694);