Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19157 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×#121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(9,12)(10,13)(14,15)(16,17)(18,25)(19,23)(20,28)(21,24)(22,26)(27,30)(29,31);(8,9,13,15,11,14,10,12);(9,14)(12,15)(16,18)(17,22)(19,26)(20,29)(21,27)(23,30)(24,31)(25,28);(16,19,21,20)(17,23,24,28)(18,26,27,29)(22,30,31,25)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12145);