Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C5×Q16 ⋊ C2 |
| Sylow | C5 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C2×C5×Q16 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | C2×Q16 | D5 |
| hypocenter | C5 | #578 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×Q16 | |
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)(6,9,11,21)(7,14,12,26)(8,16,18,34)(10,23,24,20)(13,28,25,36)(15,33,27,31)(17,35,22,19)(29,37,32,30);(6,8,11,18)(7,13,12,25)(9,19,21,35)(10,22,24,17)(14,30,26,37)(15,32,27,29)(16,23,34,20)(28,33,36,31);(7,12)(13,25)(14,26)(15,27)(28,36)(29,32)(30,37)(31,33);(2,3)(4,5)(6,7)(8,13)(9,14)(10,15)(11,12)(16,28)(17,29)(18,25)(19,30)(20,31)(21,26)(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,d^c=de,e^c=ef,e^d=ef50 conjugacy classes — view the full character table.
| maximal subgroup of | #10600061, #10600062, #10602282, … |
| maximal normal subgroup of | #10600061, #10600062, #10602282, … |
| maximal quotient of | #10610505, #10610510, #10610516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1438);