Finite Group DatabasebetaFinite Group Databasebeta| order | 30720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #6342371 ⋊ A5 |
| derived | #72209721 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | #72209721 | C2 |
| Frattini, socle | C24 | C2×#11757163 |
| Fitting | #6342371 | A5 |
| cosocle | C28 | C2×A5 |
| LCS | G ▷ #72209721 | |
| UCS | C1 | |
At least 6263 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has at least 6263 conjugacy classes of subgroups — too many to draw. Only its 7 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(4,48,60)(5,36,39)(12,15,37)(13,16,31)(14,35,38)(19,23,43)(20,24,44)(21,41,45)(22,42,46)(25,29,27)(26,49,61)(28,47,30)(50,58,62)(51,56,63)(52,54,59)(53,55,57);(4,60,29)(5,16,36)(12,37,38)(13,31,39)(14,15,35)(19,43,45)(20,44,46)(21,23,41)(22,24,42)(25,27,48)(26,61,30)(28,49,47)(50,59,54)(51,57,55)(52,62,58)(53,63,56);(4,30,28)(5,12,39)(13,38,14)(15,37,36)(16,31,35)(19,46,22)(20,45,21)(23,43,42)(24,44,41)(25,49,61)(26,29,27)(47,48,60)(50,58,57)(51,56,59)(52,54,63)(53,55,62);(3,21,20)(4,61,50)(5,12,11)(10,14,13)(17,43,42)(18,22,19)(25,28,52)(26,60,51)(27,53,47)(29,59,55)(30,57,54)(31,35,34)(33,37,36)(40,44,41)(48,62,56)(49,63,58);(2,36,10)(3,20,25)(4,17,42)(8,13,33)(9,35,11)(12,34,32)(18,19,47)(21,52,28)(22,53,27)(23,54,49)(24,55,48)(26,40,41)(29,46,56)(30,45,58)(43,50,61)(44,51,60);(1,7)(9,32)(10,33)(12,35)(14,37)(15,38)(18,40)(20,42)(22,44)(24,46)(26,47)(28,61)(30,49)(51,53)(55,56)(57,63)45 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72223000 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((4,48,60)(5,36,39)(12,15,37)(13,16,31)(14,35,38)(19,23,43)(20,24,44)(21,41,45)(22,42,46)(25,29,27)(26,49,61)(28,47,30)(50,58,62)(51,56,63)(52,54,59)(53,55,57), (4,60,29)(5,16,36)(12,37,38)(13,31,39)(14,15,35)(19,43,45)(20,44,46)(21,23,41)(22,24,42)(25,27,48)(26,61,30)(28,49,47)(50,59,54)(51,57,55)(52,62,58)(53,63,56), (4,30,28)(5,12,39)(13,38,14)(15,37,36)(16,31,35)(19,46,22)(20,45,21)(23,43,42)(24,44,41)(25,49,61)(26,29,27)(47,48,60)(50,58,57)(51,56,59)(52,54,63)(53,55,62), (3,21,20)(4,61,50)(5,12,11)(10,14,13)(17,43,42)(18,22,19)(25,28,52)(26,60,51)(27,53,47)(29,59,55)(30,57,54)(31,35,34)(33,37,36)(40,44,41)(48,62,56)(49,63,58), (2,36,10)(3,20,25)(4,17,42)(8,13,33)(9,35,11)(12,34,32)(18,19,47)(21,52,28)(22,53,27)(23,54,49)(24,55,48)(26,40,41)(29,46,56)(30,45,58)(43,50,61)(44,51,60), (1,7)(9,32)(10,33)(12,35)(14,37)(15,38)(18,40)(20,42)(22,44)(24,46)(26,47)(28,61)(30,49)(51,53)(55,56)(57,63));