Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C5 ⋊ #382 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator | C22×C5 | C22×C4 |
| Fitting | C2×C5×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C5 | C23 |
| hypercenter | C2×C4⋊C4 | D5 |
| hypocenter | C5 | #382 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
154 conjugacy classes of subgroups.
| 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)(11,13)(16,20)(18,21);(14,16,17,20)(15,18,19,21);(2,3)(4,5)(10,11,12,13)(14,15,17,19)(16,18,20,21)Polycyclic presentation:
2^6,5|a^2=e,b^2=f,c^2=d,c^a=cef,g^a=g^4,c^b=cf68 conjugacy classes — view the full character table.
| maximal subgroup of | #10597943, #10598710, #10598722, … |
| maximal normal subgroup of | #10597943, #10598710, #10598722, … |
| maximal quotient of | #10600196, #10600208, #10600340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 604);