Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#2196 (order 2560) · Inn(G) = C23×D5 · Out(G) = C4×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C5×D4 ⋊ C2 |
| Sylow | C5 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D5 |
| commutator, socle, cosocle | C2×C5 | C24 |
| Fitting | C2×C5×D4 | C2 |
| hypercenter | C2×D4 | D5 |
| hypocenter | C5 | H(5,𝔽2) |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2,4,5,3)(7,11)(10,13);(8,12)(10,13);(6,8)(7,10)(9,12)(11,13);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)Polycyclic presentation:
2^5,5|d^a=de,f^a=f^4,c^b=ce37 conjugacy classes — view the full character table.
| maximal subgroup of | #65030, #65031, #65035, … |
| maximal normal subgroup of | #65030, #65031, #65035, … |
| maximal quotient of | #65817, #65820, #65824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 219);