Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#556 ⋊ C2 |
| Sylow | C3 ⋊ #3518 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#556 | C2 |
| hypercenter | #556 | D3 |
| hypocenter | C3 | #3518 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #556 | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(4,6,7,10)(5,9,8,11)(12,16,17,15)(13,20,21,19)(14,24,25,23)(18,27,22,26);(12,15,17,16)(13,20,21,19)(14,24,25,23)(18,26,22,27);(5,8)(9,11)(12,15,17,16)(13,19,21,20)(14,23,25,24)(18,26,22,27);(2,3)(12,14,17,25)(13,18,21,22)(15,23,16,24)(19,26,20,27);(4,5)(6,9)(7,8)(10,11)(12,13,17,21)(14,22,25,18)(15,20,16,19)(23,26,24,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=f,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^b=h^2,h^e=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | #11677644, #11677648, #11677716, … |
| maximal normal subgroup of | #11677644, #11677648, #11677716, … |
| maximal quotient of | #10955384, #10955868, #10955870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19564);