Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #72216366 (order 1024) · Inn(G) = C22⋊C4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#100 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊C4 |
| commutator | C22 | C22×C4 |
| Frattini, cosocle | C2×C4 | C23 |
| prime radical | C22×D4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
93 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);(3,4,5,8,6,7,9,10);(4,7)(5,9)22 conjugacy classes — view the full character table.
| maximal subgroup of | #1390, C4×#100, #1783, … |
| maximal normal subgroup of | #1390, C4×#100, #1783, … |
| maximal quotient of | C2×#323, #1460, #1462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 92);