Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #2551 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×C4×C4⋊C4 | C2 |
| hypercenter | C4×C4⋊C4 | D3 |
| hypocenter | C3 | #2551 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(8,10,11,9);(5,7)(12,14,15,19)(13,16,17,18);(4,5,6,7)(8,9,11,10);(2,3)(9,10)(12,13,15,17)(14,18,19,16)Polycyclic presentation:
2^7,3|a^2=f,b^2=eg,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=cg78 conjugacy classes — view the full character table.
| maximal subgroup of | #10698197, #10698198, #10698202, … |
| maximal normal subgroup of | #10698197, #10698198, #10698202, … |
| maximal quotient of | #10717460, #10717463, #10718403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6291);