Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C5 ⋊ #387 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator | C22×C5 | C22×C4 |
| Fitting | C5×#95 | C2 |
| socle, prime radical, cosocle | C23×C5 | C23 |
| hypercenter | #95 | D5 |
| hypocenter | C5 | #387 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #95 | |
154 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 320 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(6,7,8,9)(16,20)(18,21);(10,11,12,13)(14,16)(15,18)(17,20)(19,21);(2,3)(4,5)(11,13)(14,15,17,19)(16,18,20,21)Polycyclic presentation:
2^6,5|a^2=f,b^2=e,c^2=d,b^a=be,g^a=g^4,c^b=cf68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#64942, #10605064, #10605066, … |
| maximal normal subgroup of | C2×#64942, #10605064, #10605066, … |
| maximal quotient of | #10599402, #10599405, #10599414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 280);