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) = C24 |
| Borel | B(4,𝔽2) |
| unitriangular | UT(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #120 |
| commutator, Frattini, cosocle | C23 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 3 generators on 8 points:
(5,8);(1,2)(3,5)(4,6)(7,8);(1,3)(2,5)(4,7)(6,8)Polycyclic presentation:
2^6|b^a=bd,c^a=ce,e^b=ef,d^c=df16 conjugacy classes — view the full character table.
| Fitting subgroup of | B(4,𝔽2)⋊C3, #73025, #73026, … |
| maximal subgroup of | #2196, #2197, #2199, … |
| maximal normal subgroup of | #2196, #2197, #2199, … |
| maximal quotient of | #1595, #1596, #1597, … |
| automorphism group of | C2×D4, #99 |
| outer automorphism group of | #490, #491, #496, … |
| holomorph of | C2×C4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 138);