Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×Q82 ⋊ C2 |
| Sylow | C3 ⋊ #3555 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×Q82 | C2 |
| prime radical | #895 | C22 |
| hypercenter | Q82 | D3 |
| hypocenter | C3 | #3555 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(20,23,24,32)(21,26,27,31)(22,30,29,35)(25,33,28,34);(2,3)(5,11)(9,18)(10,14)(17,19)(22,29)(25,28)(30,35)(33,34);(4,7,8,16)(5,10,11,14)(6,12,13,15)(9,17,18,19);(4,6,8,13)(5,9,11,18)(7,15,16,12)(10,19,14,17)(20,22)(21,25)(23,30)(24,29)(26,33)(27,28)(31,34)(32,35);(2,3)(4,5)(6,9)(7,14)(8,11)(10,16)(12,19)(13,18)(15,17)(20,21,24,27)(22,28,29,25)(23,31,32,26)(30,33,35,34)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^2,c^b=cg,d^b=df,h^d=h^263 conjugacy classes — view the full character table.
| maximal subgroup of | #10773274, #10773282, #10773289, … |
| maximal normal subgroup of | #10773274, #10773282, #10773289, … |
| maximal quotient of | #10958087, #10958208, #10958362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19624);