Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×C22⋊Q8 ⋊ C2 |
| Sylow | C5 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×C22⋊Q8 | C2 |
| prime radical | C2×#709 | C2 |
| hypercenter | C22⋊Q8 | D5 |
| hypocenter | C5 | #536 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,4,5,3)(7,9)(11,17)(15,23)(16,25)(22,24);(6,7)(8,9)(10,13,14,21)(11,16,17,25)(12,18,19,20)(15,22,23,24);(10,12,14,19)(11,15,17,23)(13,20,21,18)(16,24,25,22);(2,3)(4,5)(10,11)(12,15)(13,16)(14,17)(18,22)(19,23)(20,24)(21,25)Polycyclic presentation:
2^6,5|b^2=e,c^2=e,d^a=de,g^a=g^4,c^b=ce,d^c=df50 conjugacy classes — view the full character table.
| maximal subgroup of | #10611879, #10611886, #10611916, … |
| maximal normal subgroup of | #10611879, #10611886, #10611916, … |
| maximal quotient of | #10605602, #10605642, #10605665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1300);