Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C3×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,3)(4,5,11,17)(6,18,23,51)(7,24,28,48)(8,29,34,46)(9,16,10,15)(12,36,40,20)(13,35,43,27)(14,32,47,33)(19,52,55,67)(21,50,22,44)(25,59,54,49)(26,45,31,42)(30,39,57,38)(37,63,66,60)(41,56,65,53)(58,64,62,61);(4,6)(5,12)(7,25)(8,30)(9,21)(10,22)(11,23)(13,41)(14,44)(15,38)(16,39)(17,40)(18,29)(19,27)(20,33)(24,37)(26,56)(28,54)(31,53)(32,36)(34,57)(35,55)(42,49)(43,65)(45,59)(46,51)(47,50)(48,66)(52,58)(60,61)(62,67)(63,64);(4,7)(5,13)(6,19)(8,31)(9,35)(10,27)(11,28)(12,37)(14,45)(15,48)(16,24)(17,43)(18,49)(20,56)(21,25)(22,54)(23,55)(26,34)(29,62)(30,63)(32,61)(33,64)(36,53)(38,41)(39,65)(40,66)(42,47)(44,67)(46,58)(50,52)(51,59)(57,60)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6312);