Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#13636 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D16 |
| commutator | C16 | C22×C3×C4 |
| Frattini, cosocle | C2×C16 | C23×C3 |
| socle | C2×C3 | C2×#366 |
| prime radical | C23×C3 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ C22×C3×C16 ◁ G | |
178 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)(4,5,8,14,11,16,10,32)(6,17,21,43,19,41,23,54)(7,24,26,55,28,57,27,58)(9,30,29,40,31,59,25,15)(12,33,35,48,34,51,60,20)(13,37,38,47,39,46,56,18)(22,53,52,64,49,67,50,44)(36,45,42,61,62,65,63,66);(4,6,18,45,9,22,26,33,8,21,37,61,29,52,28,48,11,19,47,65,31,49,27,51,10,23,46,66,25,50,7,20)(5,12,24,44,15,36,56,54,32,60,58,67,59,63,39,41,16,34,57,64,40,62,38,43,14,35,55,53,30,42,13,17);(6,19)(12,34)(17,41)(20,48)(21,23)(22,49)(33,51)(35,60)(36,62)(42,63)(43,54)(44,64)(45,65)(50,52)(53,67)(61,66)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6624);