Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×#60433 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
2678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,22)(15,20)(16,18)(17,29)(19,25)(21,28)(23,36)(24,35)(26,32)(27,40)(30,31)(33,38)(34,39)(37,43)(41,42);(8,9,10,11)(13,18)(19,31)(20,32)(21,29)(22,35)(33,42)(34,40)(36,43);(12,14,16,24)(13,19,18,31)(15,23,26,37)(17,30,28,25)(20,33,32,42)(21,35,29,22)(27,41,39,38)(34,43,40,36);(12,15)(13,20)(14,23)(16,26)(17,27)(18,32)(19,33)(21,34)(22,36)(24,37)(25,38)(28,39)(29,40)(30,41)(31,42)(35,43);(15,26)(20,32)(23,37)(27,39)(33,42)(34,40)(36,43)(38,41)427 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53421);