Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#560 ⋊ C2 |
| Sylow | C3 ⋊ #3562 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#560 | C2 |
| hypercenter | #560 | D3 |
| hypocenter | C3 | #3562 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #560 | |
1050 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)(12,16,17,15)(13,21,20,19)(14,23,24,25)(18,26,22,27);(4,6,7,10)(5,9,8,11)(13,20)(14,24)(19,21)(23,25);(2,3)(4,5)(6,9)(7,8)(10,11)(12,15,17,16)(13,19,20,21)(14,23,24,25)(18,26,22,27);(5,8)(9,11)(12,14)(13,18)(15,25)(16,23)(17,24)(19,27)(20,22)(21,26);(5,8)(6,10)(12,13)(14,22)(15,21)(16,19)(17,20)(18,24)(23,26)(25,27)Polycyclic presentation:
2^7,3|c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^c=h^2,h^e=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3562, C2×#87077, #11702744, … |
| maximal normal subgroup of | D3×#3562, C2×#87077, #11702739, … |
| maximal quotient of | #10958237, #10958336, #10958480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19653);