Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C5 ⋊ #322 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C2×D5 |
| commutator | C2×C5 | C4×C8 |
| Fitting | C4×C5×C8 | C2 |
| socle, prime radical | C22×C5 | C42 |
| cosocle | C42×C5 | C22 |
| hypercenter | C4×C8 | D5 |
| hypocenter | C5 | #322 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C42 ◁ C4×C8 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4,5,3)(6,7,8,10,9,11,12,13)(15,18)(19,21);(2,3)(4,5)(14,15,16,19,17,18,20,21)Polycyclic presentation:
2^6,5|a^2=c,b^2=d,c^2=f,d^2=e,b^a=bf,g^a=g^4104 conjugacy classes — view the full character table.
| maximal subgroup of | #10598515, C8×#676, #10598840, … |
| maximal normal subgroup of | #10598515, C8×#676, #10598840, … |
| maximal quotient of | #10597933, #10597990, #10598003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 12);