Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #72224760 (order 1536) · Inn(G) = C42 · Out(G) = GL(2,ℤ/4) |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C4 | C42 |
| Frattini, cosocle | C22×C4 | C22 |
| socle | C2 | #95 |
| prime radical | C23 | Q8 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,8,6,9,5,14)(3,10,12,7,13,15,11,16);(1,3,5,12,6,13,4,11)(2,7,8,10,9,16,14,15)Polycyclic presentation:
2^6|a^2=d,b^2=e,c^2=f,d^2=f,e^2=f,b^a=bc,e^a=ef,d^b=df22 conjugacy classes — view the full character table.
| derived subgroup of | #338⋊C3, C2×#338⋊C3, #1747⋊C3, … |
| Fitting subgroup of | #338⋊C3, #67993, #67994 |
| maximal subgroup of | #1341, #1342, C2×#338, … |
| maximal normal subgroup of | #1341, #1342, C2×#338, … |
| maximal quotient of | #1275, #1291, #1292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 19);