Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Sylow | C3 ⋊ #3563 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#529 | C2 |
| prime radical | C3×#121 | C22 |
| hypercenter | #529 | D3 |
| hypocenter | C3 | #3563 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #529 | |
912 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)(5,11)(9,12)(10,17)(16,18);(6,14)(9,12)(13,19)(16,18)(20,23,24,31)(21,27,26,30)(22,29,28,34)(25,32,33,35);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(21,26)(22,28)(27,30)(29,34);(2,3)(4,6,8,14)(5,9,11,12)(7,13,15,19)(10,16,17,18)(20,22)(21,25)(23,29)(24,28)(26,33)(27,32)(30,35)(31,34);(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(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,b^2=f,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^b=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10687497, #10687501, #10687658, … |
| maximal normal subgroup of | #10687497, #10687501, #10687658, … |
| maximal quotient of | #10958140, #10958196, #10958441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18718);