Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#389 ⋊ C2 |
| Sylow | C3 ⋊ #2625 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#389 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | #389 | D3 |
| hypocenter | C3 | #2625 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #389 | |
464 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)(16,19,20,23)(17,21,18,22);(2,3)(5,7)(9,13)(12,15)(16,18,20,17)(19,21,23,22);(4,5)(6,7)(8,10,11,14)(9,12,13,15)(16,17,20,18)(19,21,23,22);(2,3)(8,9)(10,12)(11,13)(14,15)(17,18)(21,22)Polycyclic presentation:
2^7,3|b^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,h^a=h^2,c^b=cg,d^c=de,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10833997, #10834001, #10834003, … |
| maximal normal subgroup of | #10833997, #10834001, #10834003, … |
| maximal quotient of | #10721565, #10721594, #10721596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8678);