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) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4 |
| commutator | C4 | C42 |
| Frattini, cosocle | C22×C4 | C22 |
| socle | C22 | C22⋊C4 |
| prime radical | C23 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
55 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12);(6,9,11,10)Polycyclic presentation:
2^6|a^2=d,b^2=e,c^2=f,b^a=bc,c^a=cf,e^a=ef28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#339, #1733, C4×#104, … |
| maximal normal subgroup of | C2×#339, #1733, C4×#104, … |
| maximal quotient of | #1276, #1278, #1279, … |
| outer automorphism group of | #7093 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 20);