Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#3484 (order 256) · Inn(G) = D4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4 |
| commutator | C4 | C22×C4 |
| Frattini, cosocle | C2×C4 | C23 |
| socle | C22 | C22⋊C4 |
| prime radical | C2×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,2);(3,4,6,9,7,5,10,8);(3,5)(4,7)(6,8)(9,10)28 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#104, #1759, #1764, … |
| maximal normal subgroup of | C4×#104, #1759, #1764, … |
| maximal quotient of | C2×D4⋊C8, C2×Q8⋊C8, #1476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 101);