Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Sylow | C3 ⋊ #2862 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×C43 | C2 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | #2862 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C43 | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(4,6,7,5)(9,13)(12,15);(16,18,19,23)(17,20,21,22);(4,5,7,6)(8,10,11,14)(9,12,13,15);(2,3)(5,6)(8,9)(10,12)(11,13)(14,15)(16,17,19,21)(18,22,23,20)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dg,h^a=h^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #10683555, #10683557, #10683647, … |
| maximal normal subgroup of | #10683555, #10683557, #10683647, … |
| maximal quotient of | #10726379, #10726488, #10726524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5884);