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 ⋊ #3512 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#541 | C2 |
| hypercenter | #541 | D3 |
| hypocenter | C3 | #3512 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #541 | |
1032 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)(12,16,17,15)(13,20,19,25)(14,22,23,24)(18,27,21,26);(2,3)(12,15,17,16)(13,20,19,25)(14,22,23,24)(18,26,21,27);(2,3)(4,7,8,6)(5,9,10,11)(13,19)(18,21)(20,25)(26,27);(4,6,8,7)(5,9,10,11)(12,14,17,23)(13,18,19,21)(15,24,16,22)(20,27,25,26);(2,3)(4,5)(6,9)(7,11)(8,10)(12,13)(14,21)(15,20)(16,25)(17,19)(18,23)(22,27)(24,26)Polycyclic presentation:
2^7,3|b^2=g,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^2,d^b=df,h^c=h^2,h^d=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #11664303, #11664307, #11664360, … |
| maximal normal subgroup of | #11664303, #11664307, #11664360, … |
| maximal quotient of | #10955339, #10955570, #10955664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19073);