Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) = C2×#72209712 (order 3072) · Inn(G) = C23 · Out(G) = C22×#1020 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(6,8)(9,10)(11,12);(2,4)(6,8)(10,12);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)Polycyclic presentation:
2^6|b^a=bd,c^a=ce,c^b=cf22 conjugacy classes — view the full character table.
| Fitting subgroup of | #392⋊C3, #68000, #72995, … |
| maximal subgroup of | #1343, #1595, #1596, … |
| maximal normal subgroup of | #1343, #1595, #1596, … |
| maximal quotient of | #1438, #1439, #1444, … |
| outer automorphism group of | #341, #803, #1290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 73);