Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33560 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C7 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
1132 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)(10,18)(11,15)(12,16)(13,17)(14,20)(19,32)(21,33)(22,36)(23,29)(24,30)(25,31)(26,34)(27,35)(28,37)(38,39);(8,10,13,21)(9,14,17,27)(11,22,23,19)(12,20,25,35)(15,28,29,26)(16,18,31,33)(24,37,38,34)(30,36,39,32);(11,23)(12,25)(15,29)(19,22)(20,35)(26,28)(30,39)(32,36);(8,11,13,23)(9,15,17,29)(10,19,21,22)(12,24,25,38)(14,26,27,28)(16,30,31,39)(18,32,33,36)(20,34,35,37)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26548);