Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C3×#126 ⋊ C2 |
| Sylow | C3 ⋊ #3564 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C2×C3×#126 | C2 |
| prime radical | #5494 | C2 |
| hypercenter | C2×#126 | D3 |
| hypocenter | C3 | #3564 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×#126 | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(5,11)(9,12)(10,15)(18,19);(5,11)(6,14)(10,15)(13,16)(20,23,24,31)(21,27,26,30)(22,29,28,34)(25,32,33,35);(4,7,8,17)(5,10,11,15)(6,13,14,16)(9,18,12,19)(21,26)(22,28)(27,30)(29,34);(4,6)(5,9)(7,16)(8,14)(10,19)(11,12)(13,17)(15,18)(20,22)(21,25)(23,29)(24,28)(26,33)(27,32)(30,35)(31,34);(2,3)(4,5,8,11)(6,12,14,9)(7,15,17,10)(13,18,16,19)(20,21,24,26)(22,25,28,33)(23,30,31,27)(29,35,34,32)Polycyclic presentation:
2^7,3|a^2=g,c^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^2,c^b=cg,d^b=df66 conjugacy classes — view the full character table.
| maximal subgroup of | #10687533, #10687537, #10768639, … |
| maximal normal subgroup of | #10687533, #10687537, #10768639, … |
| maximal quotient of | #10958180, #10958299, #10958475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18686);