Finite Group DatabasebetaFinite Group Databasebeta| order | 23040 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C25 ⋊ S6 |
| derived | #72209614 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #72209614 | C22 |
| Frattini, socle | C24 | C2×S6 |
| Fitting | C25 | S6 |
| LCS | G ▷ #72209614 | |
| UCS | C1 | |
2455 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 2455 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(1,7)(8,23)(9,24)(11,26)(13,28)(15,30)(16,31)(18,33);(2,19)(9,24)(10,25)(11,26)(13,28)(14,29)(17,32)(18,33);(3,20)(8,23)(9,24)(12,27)(14,29)(16,31)(17,32)(18,33);(4,21)(10,25)(11,26)(12,27)(15,30)(16,31)(17,32)(18,33);(5,22)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33);(2,12,19,27)(3,8)(4,9,21,24)(10,11)(13,16,28,31)(14,30,29,15)(20,23)(25,26);(2,12,19,27)(3,8)(4,9,21,24)(10,11)(13,16,28,31)(14,30,29,15)(20,23)(25,26);(1,2,3,9)(4,12,11,10)(7,19,20,24)(8,23)(13,14,28,29)(15,17,33,31)(16,30,32,18)(21,27,26,25);(1,3,8)(2,11,4,9,25,27)(7,20,23)(10,12,19,26,21,24)(13,33,16,14,17,30)(15,28,18,31,29,32);(1,8)(2,11,19,26)(4,12)(7,23)(9,25,24,10)(13,32,28,17)(14,18,29,33)(21,27);(1,9,4)(2,10,11,12,3,8,19,25,26,27,20,23)(7,24,21)(13,31,28,16)(14,18,30)(15,29,33);(1,4,11,3,10,12)(2,9,8)(7,21,26,20,25,27)(13,33,17,29,16,30)(14,31,15,28,18,32)(19,24,23);(1,10,19,23,7,25,2,8)(3,11,21,24)(4,9,20,26)(13,18)(14,32,30,31,29,17,15,16)(28,33);(1,11,27,21,19,23,7,26,12,4,2,8)(3,10,24)(9,20,25)(13,17,16)(14,33,29,18)(28,32,31);(1,12,10,24)(2,8,3,4)(7,27,25,9)(11,26)(13,30,17,33)(14,16,29,31)(15,32,18,28)(19,23,20,21);(1,13,24,7,28,9)(2,3,14)(4,16)(8,23)(10,15,17,33,12,11)(18,27,26,25,30,32)(19,20,29)(21,31);(1,14,9)(2,3,13,19,20,28)(4,15,18,21,30,33)(7,29,24)(8,23)(10,16,12)(17,32)(25,31,27);(1,15,33,24)(2,8,3,16)(4,14,21,29)(7,30,18,9)(10,28,27,26)(11,25,13,12)(17,32)(19,23,20,31);(1,16,12)(2,9,23)(3,18,26)(4,17,29)(7,31,27)(8,19,24)(10,30,28)(11,20,33)(13,25,15)(14,21,32);(1,17,15,16,19,23,7,32,30,31,2,8)(3,18,24)(4,13,26)(9,20,33)(10,29,25,14)(11,21,28);(1,18,19,23,7,33,2,8)(3,17,16,24)(4,14,11,27,21,29,26,12)(9,20,32,31)(10,28)(13,25);(2,12,19,27)(3,8)(4,9,21,24)(10,11)(13,16,28,31)(14,30,29,15)(20,23)(25,26);(2,9)(4,12)(10,11)(13,14)(15,16)(17,18)(19,24)(21,27)(25,26)(28,29)(30,31)(32,33);(2,4,12,9,19,21,27,24)(3,11,8,10)(13,15,16,14,28,30,31,29)(17,33)(18,32)(20,26,23,25);(2,12)(3,10)(8,11)(9,24)(13,16)(14,29)(17,32)(18,33)(19,27)(20,25)(23,26)(28,31);(2,15,27)(3,18,10)(4,16,28)(8,17,11)(12,19,30)(13,21,31)(20,33,25)(23,32,26);(2,16,29,28,12,24,19,31,14,13,27,9)(3,17,10)(4,15,21,30)(8,18,11)(20,32,25)(23,33,26);(2,9)(4,12)(10,11)(13,14)(15,16)(17,18)(19,24)(21,27)(25,26)(28,29)(30,31)(32,33);(2,14,19,29)(4,16,21,31)(9,13,24,28)(10,11)(12,15,27,30)(17,18)(25,26)(32,33);(2,13,19,28)(4,15,21,30)(9,14,24,29)(12,16,27,31);(4,16,21,31)(10,33,25,18)(11,32,26,17)(12,15,27,30)42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72218858 |
| maximal normal subgroup of | C2×#72218858 |
| maximal quotient of | C2×#72218858 |
| automorphism group of | #585 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,7)(8,23)(9,24)(11,26)(13,28)(15,30)(16,31)(18,33), (2,19)(9,24)(10,25)(11,26)(13,28)(14,29)(17,32)(18,33), (3,20)(8,23)(9,24)(12,27)(14,29)(16,31)(17,32)(18,33), (4,21)(10,25)(11,26)(12,27)(15,30)(16,31)(17,32)(18,33), (5,22)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33), (2,12,19,27)(3,8)(4,9,21,24)(10,11)(13,16,28,31)(14,30,29,15)(20,23)(25,26), (2,12,19,27)(3,8)(4,9,21,24)(10,11)(13,16,28,31)(14,30,29,15)(20,23)(25,26), (1,2,3,9)(4,12,11,10)(7,19,20,24)(8,23)(13,14,28,29)(15,17,33,31)(16,30,32,18)(21,27,26,25), (1,3,8)(2,11,4,9,25,27)(7,20,23)(10,12,19,26,21,24)(13,33,16,14,17,30)(15,28,18,31,29,32), (1,8)(2,11,19,26)(4,12)(7,23)(9,25,24,10)(13,32,28,17)(14,18,29,33)(21,27), (1,9,4)(2,10,11,12,3,8,19,25,26,27,20,23)(7,24,21)(13,31,28,16)(14,18,30)(15,29,33), (1,4,11,3,10,12)(2,9,8)(7,21,26,20,25,27)(13,33,17,29,16,30)(14,31,15,28,18,32)(19,24,23), (1,10,19,23,7,25,2,8)(3,11,21,24)(4,9,20,26)(13,18)(14,32,30,31,29,17,15,16)(28,33), (1,11,27,21,19,23,7,26,12,4,2,8)(3,10,24)(9,20,25)(13,17,16)(14,33,29,18)(28,32,31), (1,12,10,24)(2,8,3,4)(7,27,25,9)(11,26)(13,30,17,33)(14,16,29,31)(15,32,18,28)(19,23,20,21), (1,13,24,7,28,9)(2,3,14)(4,16)(8,23)(10,15,17,33,12,11)(18,27,26,25,30,32)(19,20,29)(21,31), (1,14,9)(2,3,13,19,20,28)(4,15,18,21,30,33)(7,29,24)(8,23)(10,16,12)(17,32)(25,31,27), (1,15,33,24)(2,8,3,16)(4,14,21,29)(7,30,18,9)(10,28,27,26)(11,25,13,12)(17,32)(19,23,20,31), (1,16,12)(2,9,23)(3,18,26)(4,17,29)(7,31,27)(8,19,24)(10,30,28)(11,20,33)(13,25,15)(14,21,32), (1,17,15,16,19,23,7,32,30,31,2,8)(3,18,24)(4,13,26)(9,20,33)(10,29,25,14)(11,21,28), (1,18,19,23,7,33,2,8)(3,17,16,24)(4,14,11,27,21,29,26,12)(9,20,32,31)(10,28)(13,25), (2,12,19,27)(3,8)(4,9,21,24)(10,11)(13,16,28,31)(14,30,29,15)(20,23)(25,26), (2,9)(4,12)(10,11)(13,14)(15,16)(17,18)(19,24)(21,27)(25,26)(28,29)(30,31)(32,33), (2,4,12,9,19,21,27,24)(3,11,8,10)(13,15,16,14,28,30,31,29)(17,33)(18,32)(20,26,23,25), (2,12)(3,10)(8,11)(9,24)(13,16)(14,29)(17,32)(18,33)(19,27)(20,25)(23,26)(28,31), (2,15,27)(3,18,10)(4,16,28)(8,17,11)(12,19,30)(13,21,31)(20,33,25)(23,32,26), (2,16,29,28,12,24,19,31,14,13,27,9)(3,17,10)(4,15,21,30)(8,18,11)(20,32,25)(23,33,26), (2,9)(4,12)(10,11)(13,14)(15,16)(17,18)(19,24)(21,27)(25,26)(28,29)(30,31)(32,33), (2,14,19,29)(4,16,21,31)(9,13,24,28)(10,11)(12,15,27,30)(17,18)(25,26)(32,33), (2,13,19,28)(4,15,21,30)(9,14,24,29)(12,16,27,31), (4,16,21,31)(10,33,25,18)(11,32,26,17)(12,15,27,30));