Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61798 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3512 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 51 points:
(1,2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21,26,31)(22,27,35,45)(23,29,37,47)(24,30,25,40)(28,39,48,36)(32,49,33,42)(34,43,38,44)(41,50,46,51);(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22)(21,27)(23,33)(24,38)(25,34)(26,35)(28,46)(29,42)(30,44)(31,45)(32,37)(36,50)(39,51)(40,43)(41,48)(47,49);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(20,23)(21,28)(22,32)(24,36)(25,39)(26,37)(27,41)(29,40)(30,47)(31,48)(33,35)(34,50)(38,51)(42,44)(43,49)(45,46);(6,14)(9,12)(13,19)(16,18)(20,23)(21,29)(22,33)(24,39)(25,36)(26,37)(27,42)(28,40)(30,48)(31,47)(32,35)(34,50)(38,51)(41,44)(43,46)(45,49);(5,11)(9,12)(10,17)(16,18)(20,24,26,25)(21,30,31,40)(22,34,35,38)(23,36,37,39)(27,43,45,44)(28,47,48,29)(32,50,33,51)(41,49,46,42)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54786);