Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#533 ⋊ C2 |
| Sylow | C3 ⋊ #3482 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C3×#533 | C2 |
| hypercenter | #533 | D3 |
| hypocenter | C3 | #3482 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #533 | |
1094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(12,15,16,13)(14,18,19,17);(2,3)(14,19)(17,18);(5,8)(9,11)(12,13,16,15)(14,17,19,18);(2,3)(12,14)(13,17)(15,18)(16,19);(4,5)(6,9)(7,8)(10,11)(12,13,16,15)(14,18,19,17)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,e^2=fg,b^a=bf,c^a=cg,d^b=df,h^b=h^2,h^d=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10687244, #10687248, #10762166, … |
| maximal normal subgroup of | #10687244, #10687248, #10762166, … |
| maximal quotient of | #10954648, #10954650, #10954659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18854);