Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = D4×D5 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×SD16 ⋊ C2 |
| Sylow | C5 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D5 |
| commutator, cosocle | C4×C5 | C23 |
| Frattini | C4 | C22×D5 |
| Fitting | C5×SD16 | C2 |
| socle | C2×C5 | C2×D4 |
| hypercenter | SD16 | D5 |
| hypocenter | C5 | #135 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ SD16 | |
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)(6,8,10,16)(7,12,11,14)(9,18,17,15)(13,20,19,21);(7,11)(8,15)(9,17)(12,20)(14,21)(16,18);(2,3)(4,5)(6,7)(8,14)(9,13)(10,11)(12,16)(15,20)(17,19)(18,21)Polycyclic presentation:
2^5,5|c^2=e,d^2=e,b^a=be,c^a=ce,f^a=f^4,c^b=cd,d^b=de,d^c=de28 conjugacy classes — view the full character table.
| maximal subgroup of | #65736, #65737, C2×#4062, … |
| maximal normal subgroup of | #65736, #65737, C2×#4062, … |
| maximal quotient of | #65045, #65052, #65057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 137);