Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C7 · Schur multiplier (H2) = C213 |
| direct product | C2×C7×#3444 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C26×C7 |
| socle | C23×C7 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C7 ◁ G | |
8518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 6 generators on 25 points:
(1,2)(19,20,21,22,23,24,25);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18);(4,7)(8,10);(12,15)(16,18);(11,13,14,17)(12,16,15,18);(3,5,6,9)(4,8,7,10)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 55656);