Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#72210409 (order 6144) · Inn(G) = C23 · Out(G) = #11709313 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C23 | C23 |
| commutator | C22 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(7,11)(9,12)(14,17)(15,19);(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,17)(19,20);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^6|a^2=f,b^2=de,c^2=d,b^a=bd,c^a=ce28 conjugacy classes — view the full character table.
| Fitting subgroup of | #383⋊C3 |
| maximal subgroup of | #1288, #1514, #1515, … |
| maximal normal subgroup of | #1288, #1514, #1515, … |
| maximal quotient of | #1434, #1443, #1839, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 64);