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) = C28 |
| direct product | C3×#61166 |
| 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 | |
2182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(12,13,17,21)(14,18,16,20)(15,23,24,26)(19,25,27,22);(4,6)(5,8)(7,11)(9,10)(12,14,17,16)(13,18,21,20)(15,22,24,25)(19,26,27,23);(12,15)(13,19)(14,22)(16,25)(17,24)(18,26)(20,23)(21,27);(6,11)(8,10)(12,14,17,16)(13,18,21,20)(15,22,24,25)(19,26,27,23);(12,14,17,16)(13,20,21,18)(15,22,24,25)(19,23,27,26)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54154);