Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #533 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Frattini | C22 | C23×D5 |
| Fitting | C5×#117 | C2 |
| prime radical | C23×C5 | C23 |
| hypercenter | #117 | D5 |
| hypocenter | C5 | #533 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(6,8,9,13)(7,10,11,12)(15,19)(18,21);(14,16,17,20)(15,18,19,21);(14,15)(16,18)(17,19)(20,21);(2,3)(4,5)(6,7,9,11)(8,12,13,10)Polycyclic presentation:
2^6,5|a^2=e,c^2=f,d^2=e,d^a=de,g^a=g^4,d^b=df68 conjugacy classes — view the full character table.
| maximal subgroup of | #10599507, #10599523, #10599524, … |
| maximal normal subgroup of | #10599507, #10599523, #10599524, … |
| maximal quotient of | #10605115, #10605128, #10605129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1189);