Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#520 ⋊ C2 |
| Sylow | C3 ⋊ #3524 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#520 | C2 |
| hypercenter | #520 | D3 |
| hypocenter | C3 | #3524 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #520 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(12,16,17,15)(13,19,20,25)(14,23,24,22)(18,26,21,27);(12,15,17,16)(13,19,20,25)(14,23,24,22)(18,27,21,26);(2,3)(5,8)(9,11)(12,15,17,16)(13,19,20,25)(14,22,24,23)(18,26,21,27);(12,14,17,24)(13,18,20,21)(15,22,16,23)(19,26,25,27);(4,5)(6,9)(7,8)(10,11)(12,13)(14,21)(15,19)(16,25)(17,20)(18,24)(22,27)(23,26)Polycyclic presentation:
2^7,3|b^2=f,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #11677885, #11677889, #11677923, … |
| maximal normal subgroup of | #11677885, #11677889, #11677923, … |
| maximal quotient of | #10955431, #10955965, #10956060, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18474);