Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×Q32 ⋊ C2 |
| Sylow | C5 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5×D8 |
| commutator, cosocle | C5×C8 | C23 |
| Frattini | C8 | C22×D5 |
| Fitting | C5×Q32 | C2 |
| socle | C2×C5 | C2×D8 |
| hypercenter | Q32 | D5 |
| hypocenter | C5 | #508 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32 | |
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)(6,8,11,16)(7,13,12,27)(9,21,24,37)(10,26,22,19)(14,32,28,36)(15,33,25,30)(17,35,34,20)(18,31,29,23);(6,7,11,12)(8,17,16,34)(9,20,24,35)(10,25,22,15)(13,29,27,18)(14,31,28,23)(19,36,26,32)(21,33,37,30);(2,3)(4,5)(7,12)(8,16)(14,28)(15,25)(18,29)(19,26)(20,35)(21,37)Polycyclic presentation:
2^6,5|b^2=f,c^2=f,d^2=ef,e^2=f,b^a=bf,c^a=cf,g^a=g^4,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #10598683, #10598684, #10600172, … |
| maximal normal subgroup of | #10598683, #10598684, #10600172, … |
| maximal quotient of | #10599999, #10600006, #10600007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 546);