Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#541 ⋊ C2 |
| Sylow | C3 ⋊ #3455 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#541 | C2 |
| hypercenter | #541 | D3 |
| hypocenter | C3 | #3455 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #541 | |
1040 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)(12,15,16,23)(13,19,18,25)(14,22,21,27)(17,24,20,26);(2,3)(13,18)(17,20)(19,25)(24,26);(5,9)(8,11)(14,21)(17,20)(22,27)(24,26);(4,6,7,10)(5,8,9,11)(12,14)(13,17)(15,22)(16,21)(18,20)(19,24)(23,27)(25,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,20)(15,19)(16,18)(17,21)(22,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,h^a=h^2,c^b=cf,h^d=h^2,h^e=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #11663560, #11663562, #11663563, … |
| maximal normal subgroup of | #11663560, #11663562, #11663563, … |
| maximal quotient of | #10953786, #10953854, #10953947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19071);