Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#538 ⋊ C2 |
| Sylow | C3 ⋊ #3545 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#538 | C2 |
| hypercenter | #538 | D3 |
| hypocenter | C3 | #3545 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #538 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(13,18)(17,20)(19,23)(26,27);(2,3)(5,9)(8,10)(12,15,16,25)(13,19,18,23)(14,22,21,24)(17,26,20,27);(4,6,7,11)(5,8,9,10)(13,18)(14,21)(19,23)(22,24);(6,11)(8,10)(12,14)(13,17)(15,24)(16,21)(18,20)(19,27)(22,25)(23,26);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,20)(15,23)(16,18)(17,21)(19,25)(22,26)(24,27)Polycyclic presentation:
2^7,3|c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^2,c^b=cg,d^b=df,h^d=h^2,h^e=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3545, C2×#86423, #11702961, … |
| maximal normal subgroup of | D3×#3545, C2×#86423, #11702961, … |
| maximal quotient of | #10958015, #10958302, #10958303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18999);