Finite Group DatabasebetaFinite Group Databasebeta| order | 10752 |
| isoclinism | isoclinic class of #72209654 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, socle | C26 ⋊ GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Fitting, socle, cosocle | C26 | GL(3,𝔽2) |
| LCS | G | |
| UCS | C1 | |
1547 conjugacy classes of subgroups; only the 5 normal subgroups are shown.
This group has 1547 conjugacy classes of subgroups — too many to draw. Only its 5 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(2,4)(3,7)(10,13)(12,14);(2,3,5)(6,8,7)(10,14,11)(13,15,16);(1,2)(3,5)(4,6)(7,8);(1,3)(2,5)(4,8)(6,7);(1,4)(2,6)(3,8)(5,7);(9,10)(11,14)(12,15)(13,16);(9,11)(10,14)(12,13)(15,16);(9,12)(10,15)(11,13)(14,16)24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72209654 |
| maximal normal subgroup of | C2×#72209654 |
| maximal quotient of | #72209833, #72209839, #72209843, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((2,4)(3,7)(10,13)(12,14), (2,3,5)(6,8,7)(10,14,11)(13,15,16), (1,2)(3,5)(4,6)(7,8), (1,3)(2,5)(4,8)(6,7), (1,4)(2,6)(3,8)(5,7), (9,10)(11,14)(12,15)(13,16), (9,11)(10,14)(12,13)(15,16), (9,12)(10,15)(11,13)(14,16));