Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C2×C7×#2026 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C7 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ C22×C42×C7 ◁ G | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(31,32,33,34,35,36,37);(7,8,10,12)(9,13,14,11)(15,16,18,20)(17,23,24,30)(19,27,26,22)(21,28,29,25);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(16,20)(21,29)(22,27)(23,30);(4,6)(15,17,18,24)(16,21,20,29)(19,28,26,25)(22,30,27,23)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16438);