Finite Group DatabasebetaFinite Group Databasebeta| order | 7488 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | D13⋊C3×#997 |
| Fitting, derived | C13×H(5,𝔽2) ⋊ C2×C32 |
| Hall | D13×H(5,𝔽2) ⋊ C32 |
| Sylow | C13 ⋊ C2×C3×#997 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C22⋊A4×D13⋊C3 |
| commutator, Fitting, cosocle, hypocenter | C13×H(5,𝔽2) | C2×C32 |
| socle | C2×C13 | C2×C3×C22⋊A4 |
| LCS | G ▷ C13×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
876 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 3 generators on 21 points:
(2,5,3)(6,7,8)(9,10)(11,13)(12,17)(14,16)(15,19)(18,21);(2,6)(3,8)(10,12,15)(11,14,16)(13,18,19)(17,20,21);(1,7,4,5)(2,3,6,8)88 conjugacy classes — view the full character table.
| maximal subgroup of | #997×AGL(1,𝔽13), D13⋊C3×#5935, #72259510, … |
| maximal normal subgroup of | #997×AGL(1,𝔽13), D13⋊C3×#5935, #72259510, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,5,3)(6,7,8)(9,10)(11,13)(12,17)(14,16)(15,19)(18,21), (2,6)(3,8)(10,12,15)(11,14,16)(13,18,19)(17,20,21), (1,7,4,5)(2,3,6,8));