Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×#127 ⋊ C2 |
| Sylow | C5 ⋊ #453 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4083 |
| commutator, cosocle | C2×C4×C5 | C23 |
| Frattini | C2×C4 | C22×D5 |
| Fitting | C5×#127 | C2 |
| socle | C2×C5 | #120 |
| hypercenter | #127 | D5 |
| hypocenter | C5 | #453 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #127 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,4,5,3)(7,10)(8,13)(11,12);(8,13)(10,11);(2,3)(4,5)(6,7)(8,10)(9,12)(11,13)Polycyclic presentation:
2^6,5|d^2=f,c^a=cd,d^a=df,e^a=ef,g^a=g^4,c^b=ce,d^b=df,d^c=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #10597953, #10598029, #10599939, … |
| maximal normal subgroup of | #10597953, #10598029, #10599939, … |
| maximal quotient of | #10600366, #10600385, #10600386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 704);