Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1710 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #874 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×C82 | C2 |
| socle, prime radical | C22×C3 | #127 |
| hypercenter | C82 | D3 |
| hypocenter | C3 | #1710 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(5,10)(9,18)(11,15)(12,19)(20,22,24,30,23,29,31,35)(21,26,27,28,25,33,32,34);(4,6,7,13,8,14,16,17)(5,9,11,12,10,18,15,19)(21,25)(26,33)(27,32)(28,34);(2,3)(4,5,8,10)(6,12,14,19)(7,15,16,11)(9,17,18,13)(20,21)(22,28)(23,25)(24,32)(26,30)(27,31)(29,34)(33,35)Polycyclic presentation:
2^7,3|a^2=f,b^2=dg,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #10681909, #10709498, #10709499, … |
| maximal normal subgroup of | #10681909, #10709498, #10709499, … |
| maximal quotient of | #10688964, #10694932, #10694946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 640);