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×#63064 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 51 points:
(1,2,3)(4,5,9,10)(6,7,12,15)(8,11,16,17)(13,14,18,19)(20,21)(22,31)(23,24)(25,30)(26,27)(28,35)(29,36)(32,42)(33,45)(34,46)(37,38)(39,44)(40,41)(43,50)(47,49)(48,51);(5,10)(6,12)(11,17)(13,18)(21,27)(23,36)(28,31)(30,41)(32,45)(37,44)(43,46)(47,51);(4,6)(5,7)(8,13)(9,12)(10,15)(11,14)(16,18)(17,19)(20,22,26,35)(21,28,27,31)(23,32,36,45)(24,33,29,42)(25,34,40,50)(30,43,41,46)(37,47,44,51)(38,48,39,49);(4,7)(5,6)(8,14)(9,15)(10,12)(11,13)(16,19)(17,18)(20,23)(21,29)(22,32)(24,27)(25,37)(26,36)(28,42)(30,39)(31,33)(34,47)(35,45)(38,41)(40,44)(43,49)(46,48)(50,51);(4,8,9,16)(5,11,10,17)(6,13,12,18)(7,14,15,19)(20,24,26,29)(21,23,27,36)(22,33,35,42)(25,39,40,38)(28,32,31,45)(30,44,41,37)(34,49,50,48)(43,51,46,47);(20,25)(21,30)(22,34)(23,37)(24,38)(26,40)(27,41)(28,43)(29,39)(31,46)(32,47)(33,48)(35,50)(36,44)(42,49)(45,51)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56052);