Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#511790 (order 1024) · Inn(G) = C23 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C23 |
| commutator, socle | C22 | C22×C4 |
| prime radical | C23 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(5,6,7,10,8,9,11,12);(1,2,3,4);(6,9)(10,12)Polycyclic presentation:
2^6|a^2=e,b^2=d,e^2=f,b^a=bd,c^a=cf28 conjugacy classes — view the full character table.
| derived subgroup of | #94658, #94659, #94660, … |
| Frattini subgroup of | #7356, #7357, #7360, … |
| maximal subgroup of | #1286, #1292, #1299, … |
| maximal normal subgroup of | #1286, #1292, #1299, … |
| maximal quotient of | #1566, #1567, #1568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 104);