Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = D4×AGL(1,𝔽5)×S4 (order 3840) · Inn(G) = C23×D5 · Out(G) = C2×C4×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#41 ⋊ C2 |
| Sylow | C5 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D5 |
| commutator, socle, cosocle | C2×C5 | C24 |
| Fitting | C5×#41 | C2 |
| hypercenter | #41 | D5 |
| hypocenter | C5 | 2−1+4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(6,10,11,9)(7,13,14,18)(8,15,16,17)(12,20,21,19);(6,9,11,10)(7,13,14,18)(8,15,16,17)(12,19,21,20);(6,8,11,16)(7,12,14,21)(9,17,10,15)(13,20,18,19);(2,3)(4,5)(6,7)(8,12)(9,13)(10,18)(11,14)(15,19)(16,21)(17,20)Polycyclic presentation:
2^5,5|b^2=e,c^2=e,d^2=e,d^a=de,f^a=f^4,c^b=ce37 conjugacy classes — view the full character table.
| maximal subgroup of | #65115, #65116, #65490, … |
| maximal normal subgroup of | #65115, #65116, #65490, … |
| maximal quotient of | #65850, #65852, #65853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 225);