Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63063 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 43 points:
(1,2,3)(4,5,8,9)(6,7,10,11)(12,13,18,22)(14,23,24,19)(15,20,28,34)(16,17,21,31)(25,36,38,32)(26,37,33,27)(29,30,35,41)(39,43,42,40);(5,9)(6,10)(14,24)(15,28)(19,23)(20,34)(26,33)(27,37)(29,35)(30,41);(4,6)(5,7)(8,10)(9,11)(12,14,18,24)(13,19,22,23)(15,25,28,38)(16,26,21,33)(17,27,31,37)(20,32,34,36)(29,39,35,42)(30,40,41,43);(4,7)(5,6)(8,11)(9,10)(12,15)(13,20)(14,25)(16,29)(17,30)(18,28)(19,32)(21,35)(22,34)(23,36)(24,38)(26,39)(27,40)(31,41)(33,42)(37,43);(12,16)(13,17)(14,26)(15,29)(18,21)(19,27)(20,30)(22,31)(23,37)(24,33)(25,39)(28,35)(32,40)(34,41)(36,43)(38,42);(12,17)(13,21)(14,27)(15,30)(16,22)(18,31)(19,33)(20,35)(23,26)(24,37)(25,40)(28,41)(29,34)(32,42)(36,39)(38,43)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56051);