Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #64945 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #398 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator, socle, prime radical, cosocle | C23×C5 | C23 |
| Fitting | C2×C5×C4⋊C4 | C2 |
| hypercenter | C2×C4⋊C4 | D5 |
| hypocenter | C5 | #398 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
150 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 63 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)(7,9)(10,12,13,11);(11,12)(14,16,17,21)(15,18,19,20);(2,3)(4,5)(6,7)(8,9)(10,11,13,12)(14,15,17,19)(16,20,21,18)Polycyclic presentation:
2^6,5|a^2=e,b^2=ef,c^2=f,b^a=be,c^a=cd,g^a=g^4,c^b=cf62 conjugacy classes — view the full character table.
| maximal subgroup of | #10605400, #10605401, #10605402, … |
| maximal normal subgroup of | #10605400, #10605401, #10605402, … |
| maximal quotient of | #10600203, #10600204, #10600207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 611);