Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#135 ⋊ C2 |
| Sylow | C5 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×#135 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | #135 | D5 |
| hypocenter | C5 | #578 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #135 | |
248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,4,5,3)(6,9,11,19)(7,14,12,26)(8,17,16,34)(10,23,24,21)(13,28,25,36)(15,33,27,31)(18,35,22,20)(29,37,32,30);(6,8,11,16)(7,13,12,25)(9,20,19,35)(10,22,24,18)(14,30,26,37)(15,32,27,29)(17,23,34,21)(28,33,36,31);(7,12)(8,16)(9,19)(15,27)(18,22)(21,23)(28,36)(30,37);(2,3)(4,5)(6,7)(8,13)(9,14)(10,15)(11,12)(16,25)(17,28)(18,29)(19,26)(20,30)(21,31)(22,32)(23,33)(24,27)(34,36)(35,37)Polycyclic presentation:
2^6,5|c^2=f,d^2=f,e^2=f,b^a=bf,g^a=g^4,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef50 conjugacy classes — view the full character table.
| maximal subgroup of | #10600036, #10600038, #10602311, … |
| maximal normal subgroup of | #10600036, #10600038, #10602311, … |
| maximal quotient of | #10610552, #10610554, #10610559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1443);