Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| isoclinism | isoclinic class of #72209596 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C24 ⋊ GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | AGL(3,𝔽2) |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | C24 | GL(3,𝔽2) |
| LCS | G | |
| UCS | C1 ◁ C2 | |
129 conjugacy classes of subgroups; only the 4 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 4 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(1,2)(3,6)(4,5)(7,10)(8,11)(9,13)(12,15)(14,16);(1,3,2)(4,7,5)(6,8,9)(10,12,14);(1,4)(2,5)(6,10)(9,14);(1,4)(3,7)(6,10)(8,12);(6,10)(8,12)(9,14)(11,15);(1,4)(2,5)(3,7)(6,10)(8,12)(9,14)(11,15)(13,16)16 conjugacy classes — view the full character table.
| maximal subgroup of | #72209775, #72209833, #72209839, … |
| maximal normal subgroup of | C2×#72209596 |
| maximal quotient of | #72209612, #72209775, #72209793, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,2)(3,6)(4,5)(7,10)(8,11)(9,13)(12,15)(14,16), (1,3,2)(4,7,5)(6,8,9)(10,12,14), (1,4)(2,5)(6,10)(9,14), (1,4)(3,7)(6,10)(8,12), (6,10)(8,12)(9,14)(11,15), (1,4)(2,5)(3,7)(6,10)(8,12)(9,14)(11,15)(13,16));