Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #14679 (order 256) · Inn(G) = C23 · Out(G) = C2×C22⋊C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
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 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^6|a^2=e,c^2=d,e^2=f,b^a=bd,c^b=cf28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#433, #2921, #2922, … |
| maximal normal subgroup of | C2×#433, #2921, #2922, … |
| maximal quotient of | #1746, #1750, #1768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 114);