Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×D4×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4×D4 ◁ G | |
3230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2,3)(4,5,6,7);(5,7);(8,9)(10,12)(11,13)(14,15);(9,12)(10,15)(13,14);(9,13)(12,14)165 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×D4×AGL(1,ℤ/8), #72253889 |
| maximal subgroup of | D3×D4×AGL(1,ℤ/8), #72253889, C3×#10576108, … |
| maximal normal subgroup of | D3×D4×AGL(1,ℤ/8), #72253889, C3×#10576108 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54053);