Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2521 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#124 | C2 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | C2×#124 | D3 |
| hypocenter | C3 | #2521 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
518 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)(4,6,7,5)(17,21)(20,22);(10,14)(12,15)(17,21)(18,23);(2,3)(4,5,7,6)(8,10)(9,12)(11,14)(13,15)(16,18,19,23)(17,20,21,22);(5,6)(8,9,11,13)(10,12,14,15)(16,17,19,21)(18,22,23,20)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,d^2=ef,b^a=be,c^a=cf,d^a=de,c^b=cg,h^b=h^2,h^d=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2521, C2×#82968, #10964333, … |
| maximal normal subgroup of | D3×#2521, C2×#82968, #10964333, … |
| maximal quotient of | #10717284, #10717498, #10718457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15544);