Finite Group DatabasebetaFinite Group Databasebeta| order | 5376 |
| isoclinism | isoclinic class of #72209595 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| direct product | C2×#72209595 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | AGL(3,𝔽2) |
| commutator, hypocenter | #72209595 | C2 |
| Frattini | C2 | C2×AGL(3,𝔽2) |
| Fitting, socle | C25 | GL(3,𝔽2) |
| cosocle | C24 | C2×GL(3,𝔽2) |
| LCS | G ▷ #72209595 | |
| UCS | C1 ◁ C22 | |
455 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2)(4,7)(6,10)(11,14,13,18)(12,17,16,24)(15,20,23,19)(21,22,25,26);(3,6,7,8,9,5,4)(12,15,17,21,20,22,14)(16,23,24,25,19,26,18)38 conjugacy classes — view the full character table.
| maximal subgroup of | #72227627, #72227690, #72227747, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,7)(6,10)(11,14,13,18)(12,17,16,24)(15,20,23,19)(21,22,25,26), (3,6,7,8,9,5,4)(12,15,17,21,20,22,14)(16,23,24,25,19,26,18));