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 | C4×C5×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)(14,15,16,19,17,18,20,21);(6,8,9,13)(7,10,11,12);(2,3)(4,5)(6,7,9,11)(8,12,13,10)(15,18)(19,21)Polycyclic presentation:
2^6,5|a^2=e,b^2=e,c^2=d,d^2=f,b^a=be,c^a=cf,g^a=g^492 conjugacy classes — view the full character table.
| maximal subgroup of | #10598009, #10598050, #10598848, … |
| maximal normal subgroup of | #10598009, #10598050, #10598848, … |
| maximal quotient of | #10599406, #10599407, #10599411, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 306);