Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) = C2×#1020⋊C2 (order 384) · Inn(G) = #120 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #120 |
| commutator, Frattini, cosocle | C23 | C23 |
| prime radical | H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)Polycyclic presentation:
2^6|a^2=f,b^a=bd,c^a=ce,e^b=ef,d^c=df16 conjugacy classes — view the full character table.
| Fitting subgroup of | #458⋊C3, #73029, #73030, … |
| maximal subgroup of | #2197, #2198, #2200, … |
| maximal normal subgroup of | #2197, #2198, #2200, … |
| maximal quotient of | #1607, #1608, #1609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 139);