Finite Group DatabasebetaFinite Group Databasebeta| order | 164 |
| isoclinism | isoclinic class of #4223 |
| automorphism | Aut(G) = AGL(1,𝔽41) (order 1640) · Inn(G) = G · Out(G) = C2×C5 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C41 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C41 | C4 |
| prime radical, cosocle | D41 | C2 |
| LCS | G ▷ C41 | |
| UCS | C1 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 41 |
A minimal faithful permutation representation — 2 generators on 41 points:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,35)(23,33,34,40)(27,37,38,36)(29,41,39,32);(1,2,5,10,19,30,34,22,13,6,11,20,32,33,17,29,38,26,15,24,36,37,21,8,16,27,39,28,40,41,25,12,3,7,14,23,35,31,18,9,4)Polycyclic presentation:
2^2,41|a^2=b,c^a=c^32,c^b=c^4014 conjugacy classes — view the full character table.
| maximal subgroup of | #66502, C2×#4223, C3×#4223, … |
| maximal normal subgroup of | #66502, C2×#4223, C3×#4223, … |
| maximal quotient of | #66493, C2×#4223, C3×#4223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(164, 3);