Finite Group DatabasebetaFinite Group Databasebeta| order | 1696 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | #124×C53 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C53 |
| prime radical | C2×D4×C53 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 3 generators on 65 points:
(2,3)(7,11)(9,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65);(1,2,4,3)(5,6,8,10)(7,9,11,12);(1,3,4,2)(5,7)(6,9)(8,11)(10,12)742 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1696, 31);