Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#73026 (order 7680) · Inn(G) = C23×D5 · Out(G) = C4×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C5×Q8 ⋊ C2 |
| Sylow | C5 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D5 |
| commutator, socle, cosocle | C2×C5 | C24 |
| Fitting | C2×C5×Q8 | C2 |
| hypercenter | C2×Q8 | D5 |
| hypocenter | C5 | 2−1+4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
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)(7,13)(11,19)(12,21)(18,20);(6,9,10,17)(7,12,13,21)(8,14,15,16)(11,18,19,20);(6,8,10,15)(7,11,13,19)(9,16,17,14)(12,20,21,18);(2,3)(4,5)(6,7)(8,11)(9,12)(10,13)(14,18)(15,19)(16,20)(17,21)Polycyclic presentation:
2^5,5|b^2=e,c^2=e,d^a=de,f^a=f^4,c^b=ce37 conjugacy classes — view the full character table.
| maximal subgroup of | #65041, #65042, #65351, … |
| maximal normal subgroup of | #65041, #65042, #65351, … |
| maximal quotient of | #65834, #65838, #65841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 222);