Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25448 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2428 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 67 points:
(1,2,3)(4,5,11,15,10,17,9,16)(6,18,32,37,22,36,21,12)(7,23,26,13,28,43,27,40)(8,29,33,42,31,41,30,14)(19,38,47,44,24,34,48,52)(20,49,59,61,51,35,50,45)(25,55,56,53,58,39,57,63)(46,65,66,62,54,64,67,60);(4,6,10,22)(5,12)(7,24)(8,20,31,51)(9,21,11,32)(13,38,40,34)(14,35)(15,18)(16,36)(17,37)(19,28)(23,52,43,44)(25,54)(26,48)(27,47)(29,45)(30,50,33,59)(39,62,55,60)(41,61)(42,49)(46,58)(53,65,63,64)(56,67)(57,66);(4,7)(5,13)(6,19)(8,25)(9,26)(10,28)(11,27)(12,34)(14,39)(15,23)(16,43)(17,40)(18,44)(20,46)(21,47)(22,24)(29,53)(30,56)(31,58)(32,48)(33,57)(35,60)(36,52)(37,38)(41,63)(42,55)(45,64)(49,62)(50,66)(51,54)(59,67)(61,65);(4,8,11,33,10,31,9,30)(5,14,16,41,17,42,15,29)(6,20,32,59,22,51,21,50)(7,25,27,57,28,58,26,56)(12,35,36,61,37,49,18,45)(13,39,43,63,40,55,23,53)(19,46,48,67,24,54,47,66)(34,60,52,65,38,62,44,64)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18436);