Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×H(5,𝔽2) ⋊ C2 |
| Sylow | C5 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C22⋊D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×H(5,𝔽2) | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | H(5,𝔽2) | D5 |
| hypocenter | C5 | #576 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ H(5,𝔽2) | |
268 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(7,12)(9,17)(11,19)(13,14)(16,21)(18,20);(8,15)(11,18)(16,21)(19,20);(6,8)(7,11)(9,16)(10,15)(12,19)(13,20)(14,18)(17,21);(2,3)(4,5)(6,7)(8,11)(9,12)(10,14)(13,17)(15,18)(16,19)(20,21)Polycyclic presentation:
2^6,5|e^2=f,d^a=de,e^a=ef,g^a=g^4,c^b=cf,e^d=ef56 conjugacy classes — view the full character table.
| maximal subgroup of | #10612011, #10612013, #10612025, … |
| maximal normal subgroup of | #10612011, #10612013, #10612025, … |
| maximal quotient of | #10614268, #10614269, #10614270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1507);