Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C5×2−1+4 ⋊ C2 |
| Sylow | C5 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×D5 |
| commutator, socle, cosocle | C2×C5 | C25 |
| Fitting | C5×2−1+4 | C2 |
| hypercenter | 2−1+4 | D5 |
| hypocenter | C5 | #585 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2,4,5,3)(6,9,10,16)(7,13,12,20)(8,14,15,17)(11,19,18,21);(7,12)(11,18)(13,20)(19,21);(6,8,10,15)(7,11,12,18)(9,17,16,14)(13,21,20,19);(6,7)(8,11)(9,13)(10,12)(14,19)(15,18)(16,20)(17,21);(2,3)(4,5)(9,16)(13,20)(14,17)(19,21)Polycyclic presentation:
2^6,5|c^2=f,e^2=f,e^a=ef,g^a=g^4,d^b=df,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #10605020, #10612015, #10612017, … |
| maximal normal subgroup of | #10605020, #10612015, #10612017, … |
| maximal quotient of | #10617773, #10617774, #10617775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1625);