Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#397 ⋊ C2 |
| Sylow | C3 ⋊ #2715 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#397 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #397 | D3 |
| hypocenter | C3 | #2715 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #397 | |
498 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)(9,13)(12,14)(16,18,19,23)(17,21,20,22);(2,3)(5,7)(8,10,11,15)(9,12,13,14)(16,17)(18,21)(19,20)(22,23);(2,3)(4,5)(6,7)(16,17,19,20)(18,22,23,21);(8,9)(10,14)(11,13)(12,15)(17,20)(21,22)Polycyclic presentation:
2^7,3|b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^b=h^2,h^c=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #11002566, #11002573, #11002668, … |
| maximal normal subgroup of | #11002566, #11002573, #11002668, … |
| maximal quotient of | #10722795, #10722885, #10723029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10163);