Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×C4×#41 ⋊ C2 |
| Sylow | C3 ⋊ #3515 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×C4×#41 | C2 |
| hypercenter | C4×#41 | D3 |
| hypocenter | C3 | #3515 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×#41 | |
1014 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)(13,19)(17,20)(18,25)(24,26);(14,22)(17,20)(21,27)(24,26);(5,8)(9,11)(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26);(12,14)(13,17)(15,21)(16,22)(18,24)(19,20)(23,27)(25,26);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13,16,19)(14,20,22,17)(15,18,23,25)(21,26,27,24)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,h^a=h^2,d^b=df84 conjugacy classes — view the full character table.
| maximal subgroup of | #10687060, #10687064, #10763178, … |
| maximal normal subgroup of | #10687060, #10687064, #10763178, … |
| maximal quotient of | #10955371, #10955847, #10955878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18407);