Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #3958 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×#135 ⋊ C2 |
| Sylow | C5 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #686 |
| commutator, cosocle | C5×C8 | C23 |
| Frattini | C8 | C22×D5 |
| Fitting | C5×#135 | C2 |
| socle | C2×C5 | C2×D8 |
| hypercenter | #135 | D5 |
| hypocenter | C5 | #508 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #135 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,4,5,3)(7,13)(9,20)(10,21)(12,24)(14,15)(16,29)(17,30)(18,31)(22,34)(23,28)(25,26)(27,36)(32,37)(33,35);(6,8,11,19)(7,12,16,27)(9,17,22,32)(10,18,21,31)(13,24,29,36)(14,25,28,35)(15,26,23,33)(20,30,34,37);(2,3)(4,5)(6,7)(8,12)(9,13)(10,23)(11,16)(14,20)(15,21)(17,24)(18,33)(19,27)(22,29)(25,30)(26,31)(28,34)(32,36)(35,37)Polycyclic presentation:
2^6,5|b^2=f,d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^4,d^c=de,e^c=ef50 conjugacy classes — view the full character table.
| maximal subgroup of | #10600031, #10600032, #10601714, … |
| maximal normal subgroup of | #10600031, #10600032, #10601714, … |
| maximal quotient of | #10601441, #10601444, #10601446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 821);