Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×#99 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C2×Q8 | C24×C7 |
| prime radical | C7×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C7×Q8 ◁ C2×C7×H(5,𝔽2) ◁ G | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,14,19,15,20,11,17)(10,21,22,37,24,32,29,39)(12,18,26,36,13,27,25,35)(16,30,33,28,34,38,31,23);(8,10)(9,16)(11,22)(12,23)(13,28)(14,29)(15,24)(17,31)(18,32)(19,33)(20,34)(21,27)(25,30)(26,38)(35,37)(36,39);(8,11,15,14)(9,17,20,19)(10,22,24,29)(12,26,13,25)(16,31,34,33)(18,36,27,35)(21,37,32,39)(23,38,28,30);(8,12,15,13)(9,18,20,27)(10,23,24,28)(11,25,14,26)(16,32,34,21)(17,35,19,36)(22,30,29,38)(31,37,33,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26359);