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×#63067 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 6 generators on 51 points:
(1,2,3)(4,5,9,7)(6,12,14,10)(8,15,16,11)(13,17,19,18)(20,21,26,30)(22,31,32,27)(23,28,36,42)(24,25,29,39)(33,44,46,40)(34,45,41,35)(37,38,43,49)(47,51,50,48);(4,6,9,14)(5,10,7,12)(8,13,16,19)(11,17,15,18)(22,32)(23,36)(27,31)(28,42)(34,41)(35,45)(37,43)(38,49);(4,7,9,5)(6,10,14,12)(8,15,16,11)(13,17,19,18)(20,22,26,32)(21,27,30,31)(23,33,36,46)(24,34,29,41)(25,35,39,45)(28,40,42,44)(37,47,43,50)(38,48,49,51);(4,8,9,16)(5,11,7,15)(6,13,14,19)(10,17,12,18)(20,23)(21,28)(22,33)(24,37)(25,38)(26,36)(27,40)(29,43)(30,42)(31,44)(32,46)(34,47)(35,48)(39,49)(41,50)(45,51);(20,24)(21,25)(22,34)(23,37)(26,29)(27,35)(28,38)(30,39)(31,45)(32,41)(33,47)(36,43)(40,48)(42,49)(44,51)(46,50);(20,25)(21,29)(22,35)(23,38)(24,30)(26,39)(27,41)(28,43)(31,34)(32,45)(33,48)(36,49)(37,42)(40,50)(44,47)(46,51)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56055);