Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C5×#41 ⋊ C2 |
| Sylow | C5 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×D5 |
| commutator, socle, cosocle | C2×C5 | C25 |
| Frattini | C2 | C24×D5 |
| Fitting | C2×C5×#41 | C2 |
| hypercenter | C2×#41 | D5 |
| hypocenter | C5 | #585 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C2×#41 | |
810 conjugacy classes of subgroups; only the 443 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 443 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(7,13)(11,18)(12,19)(17,21);(8,15)(11,18)(14,20)(17,21);(6,9,10,16)(7,12,13,19)(8,14,15,20)(11,17,18,21);(6,8)(7,11)(9,14)(10,15)(12,17)(13,18)(16,20)(19,21);(2,3)(4,5)(6,7)(8,11)(9,12)(10,13)(14,17)(15,18)(16,19)(20,21)Polycyclic presentation:
2^6,5|c^2=f,e^a=ef,g^a=g^4,d^b=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #10600802, #10600803, #10600847, … |
| maximal normal subgroup of | #10600802, #10600803, #10600847, … |
| maximal quotient of | #10617626, #10617628, #10617632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1621);