Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5 ⋊ #446 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D5 |
| commutator, socle, prime radical | C22×C5 | C22×C4 |
| Fitting | C5×C4⋊C8 | C2 |
| cosocle | C2×C4×C5 | C23 |
| hypercenter | C4⋊C8 | D5 |
| hypocenter | C5 | #446 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
94 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)(8,12)(10,13)(14,16,17,19,18,20,21,15);(6,8,9,12)(7,10,11,13);(2,3)(4,5)(6,7,9,11)(8,10,12,13)(14,15,17,16,18,19,21,20)Polycyclic presentation:
2^6,5|a^2=e,b^2=d,c^2=de,e^2=f,c^a=cf,g^a=g^4,c^b=cd68 conjugacy classes — view the full character table.
| maximal subgroup of | #10598582, #10598947, #10598949, … |
| maximal normal subgroup of | #10598582, #10598947, #10598949, … |
| maximal quotient of | #10599405, #10599413, #10599446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 458);