Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| isoclinism | isoclinic class of #72209595 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(3,𝔽2) |
| commutator, hypocenter | G | C1 |
| Fitting, socle, cosocle | C24 | GL(3,𝔽2) |
| LCS | G | |
| UCS | C1 ◁ C2 | |
116 conjugacy classes of subgroups.
Permutation generators:
(2,4)(3,7)(9,10,13,11)(12,15,14,16)(17,21,19,22)(18,23,20,24);(2,3,5)(6,8,7)(9,12,10)(11,13,14)(15,17,18)(16,19,20);(9,13)(10,11)(12,14)(15,16)(17,19)(18,20)(21,22)(23,24);(1,2)(3,5)(4,6)(7,8);(1,3)(2,5)(4,8)(6,7);(1,4)(2,6)(3,8)(5,7)19 conjugacy classes — view the full character table.
| derived subgroup of | #72227541, C2×#72209595 |
| maximal subgroup of | #72209765, #72209775, #72209804, … |
| maximal normal subgroup of | #72227541, C2×#72209595 |
| maximal quotient of | #72209612, #72209765, #72209785, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((2,4)(3,7)(9,10,13,11)(12,15,14,16)(17,21,19,22)(18,23,20,24), (2,3,5)(6,8,7)(9,12,10)(11,13,14)(15,17,18)(16,19,20), (9,13)(10,11)(12,14)(15,16)(17,19)(18,20)(21,22)(23,24), (1,2)(3,5)(4,6)(7,8), (1,3)(2,5)(4,8)(6,7), (1,4)(2,6)(3,8)(5,7));