Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = C2×B(4,𝔽2) (order 128) · Inn(G) = #99 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #99 |
| commutator | C23 | C2×C4 |
| Frattini, prime radical, cosocle | C2×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,9)(4,12,13,7)(5,8)(6,10,14,16)(11,15);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)Polycyclic presentation:
2^6|a^2=d,b^2=f,b^a=bc,c^a=ce,e^a=ef,d^b=de,d^c=df13 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#352, #2120, #2121, … |
| maximal normal subgroup of | C2×#352, #2120, #2121, … |
| maximal quotient of | #1317, #1326, #1344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 33);