Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10336 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#10336 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #392 |
| commutator, Frattini, cosocle | C25 | C23×C7 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C25×C7 ◁ G | |
1566 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17)(18,20)(19,21)(22,23);(12,15)(16,18)(17,20)(19,22)(21,23);(8,10)(9,12)(11,14)(13,15)(20,23)238 conjugacy classes — view the full character table.
| Fitting subgroup of | D7×#10336, ΓL(1,𝔽8)×#10336, AGL(1,𝔽7)×#10336 |
| maximal subgroup of | D7×#10336, ΓL(1,𝔽8)×#10336 |
| maximal normal subgroup of | D7×#10336, ΓL(1,𝔽8)×#10336 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3324);