Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C5×M16 ⋊ C2 |
| Sylow | C5 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D20 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C2×C5×M16 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | C2×M16 | D5 |
| hypocenter | C5 | #575 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×M16 | |
262 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 262 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)(6,9,10,18,11,17,8,15)(7,13,12,16,14,20,19,21);(6,8,11,10)(7,12,14,19)(9,15,17,18)(13,16,20,21);(9,17)(13,20)(15,18)(16,21);(2,3)(4,5)(6,7)(8,12)(9,16)(10,19)(11,14)(13,18)(15,20)(17,21)Polycyclic presentation:
2^6,5|c^2=f,d^2=e,e^2=f,d^a=de,e^a=ef,g^a=g^4,d^b=df62 conjugacy classes — view the full character table.
| maximal subgroup of | #10599491, #10599496, #10599512, … |
| maximal normal subgroup of | #10599491, #10599496, #10599512, … |
| maximal quotient of | #10610210, #10610225, #10610228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1420);