Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#26851 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,11)(7,9)(8,10)(12,13)(14,26)(15,32)(16,36)(17,24)(18,23)(19,25)(20,29)(21,34)(22,27)(28,42)(30,39)(31,38)(33,40)(35,41)(37,43);(4,6,8,7)(5,9,10,11)(12,14,18,30,19,31,17,29)(13,20,24,38,25,39,23,26)(15,33,34,27,35,16,28,37)(21,40,32,43,42,36,41,22);(4,7,8,6)(5,9,10,11)(12,15,17,28,19,35,18,34)(13,21,23,41,25,42,24,32)(14,27,29,33,31,37,30,16)(20,22,26,36,39,43,38,40);(5,10)(9,11)(12,16,17,27,19,33,18,37)(13,22,23,36,25,43,24,40)(14,28,29,35,31,34,30,15)(20,21,26,41,39,42,38,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 19839);