Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#525 (order 128) · Inn(G) = C22 · Out(G) = C2×#41 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C22 |
| commutator | C2 | C2×C16 |
| socle, prime radical | C22 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
25 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,4)(5,6,7,10,8,11,13,16,9,12,14,17,15,18,19,20);(1,2,3,4)Polycyclic presentation:
2^6|a^2=d,b^2=c,d^2=e,e^2=f,b^a=bc40 conjugacy classes — view the full character table.
| maximal subgroup of | D4⋊C16, #1330, Q8⋊C16, … |
| maximal normal subgroup of | D4⋊C16, #1330, Q8⋊C16, … |
| maximal quotient of | #1367, #1370, #1374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 44);