Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C5×#41 ⋊ C2 |
| Sylow | C5 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×C22⋊D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C2×C5×#41 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | C2×#41 | D5 |
| hypocenter | C5 | #575 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#41 | |
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)(7,13)(8,15)(10,19)(11,17)(12,20)(14,21)(16,18);(6,8,11,10)(7,12,14,18)(9,15,17,19)(13,20,21,16);(9,17)(13,21)(15,19)(16,20);(2,3)(4,5)(6,7)(8,12)(9,16)(10,18)(11,14)(13,15)(17,20)(19,21)Polycyclic presentation:
2^6,5|c^2=f,e^2=f,d^a=de,e^a=ef,g^a=g^4,d^b=df,e^d=ef62 conjugacy classes — view the full character table.
| maximal subgroup of | #10600805, #10600806, #10600836, … |
| maximal normal subgroup of | #10600805, #10600806, #10600836, … |
| maximal quotient of | #10613938, #10613944, #10613946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1494);