Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4⋊C4×Dic5 |
| Sylow | C5 ⋊ C4×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator | C2×C5 | C2×C42 |
| Fitting | C2×C5×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C5 | C23 |
| hypercenter | C2×C4⋊C4 | D5 |
| hypocenter | C5 | C4×C4⋊C4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
194 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,3)(4,5)(10,11,12,13);(1,2,4,5,3)(14,15,16,17);(6,7,8,9)(15,17)80 conjugacy classes — view the full character table.
| maximal subgroup of | #10598721, #10599573, #10599574, … |
| maximal normal subgroup of | #10598721, #10599573, #10599574, … |
| maximal quotient of | Dic5×#95, #10600194, #10600200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 602);