Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #677 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×AGL(1,ℤ/8) (order 1280) · Inn(G) = D20 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C5×C8 ⋊ C2 |
| Sylow | C5 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D20 |
| commutator, cosocle | C4×C5 | C23 |
| Frattini | C4 | C22×D5 |
| Fitting | C2×C5×C8 | C2 |
| socle | C2×C5 | C2×D4 |
| hypercenter | C2×C8 | D5 |
| hypocenter | C5 | #135 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(6,9,10,17,11,18,8,15)(7,13,12,16,14,20,19,21);(6,8,11,10)(7,12,14,19)(9,15,18,17)(13,16,20,21);(2,3)(4,5)(6,7)(8,12)(9,16)(10,19)(11,14)(13,17)(15,20)(18,21)Polycyclic presentation:
2^5,5|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^446 conjugacy classes — view the full character table.
| maximal subgroup of | #64708, #64717, #64730, … |
| maximal normal subgroup of | #64708, #64717, #64730, … |
| maximal quotient of | #64972, C4×#677, C4×D40, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 125);