Finite Group DatabasebetaFinite Group Databasebeta| order | 7200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| socle, derived | A52 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | A52 | C2 |
| LCS | G ▷ A52 | |
| UCS | C1 | |
246 conjugacy classes of subgroups; only the 5 normal subgroups are shown.
This group has 246 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:
(2,5)(3,4)(6,9)(7,11)(8,13)(10,12)(14,23)(15,19)(16,22)(17,21)(18,24)(20,25);(1,2)(3,9)(4,11)(5,13)(6,14)(7,16)(10,17)(12,15)(18,19)(20,24)(21,25)(22,23);(1,3)(2,6)(4,12)(5,11)(7,17)(8,16)(9,19)(13,23)(14,24)(15,21)(18,20)(22,25);(1,4)(2,7)(3,10)(5,9)(6,15)(8,14)(11,21)(13,22)(16,20)(17,19)(18,23)(24,25);(1,3)(2,6)(5,11)(7,9)(8,16)(10,12)(13,23)(14,22)(15,19)(17,21)(18,24)(20,25);(1,2)(3,9)(4,11)(6,7)(8,13)(10,17)(12,15)(14,23)(16,22)(18,24)(19,21)(20,25);(2,8,5,13)(3,4,10,12)(6,14,15,24)(7,18,17,23)(9,20,19,16)(11,22,21,25)26 conjugacy classes — view the full character table.
| maximal subgroup of | #72219552, S52, C2×#72219548, … |
| maximal normal subgroup of | #72219552, S52, C2×#72219548, … |
| maximal quotient of | SO+(4,𝔽5), C2×#72219548, C3×#72219548, … |
| holomorph of | A5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((2,5)(3,4)(6,9)(7,11)(8,13)(10,12)(14,23)(15,19)(16,22)(17,21)(18,24)(20,25), (1,2)(3,9)(4,11)(5,13)(6,14)(7,16)(10,17)(12,15)(18,19)(20,24)(21,25)(22,23), (1,3)(2,6)(4,12)(5,11)(7,17)(8,16)(9,19)(13,23)(14,24)(15,21)(18,20)(22,25), (1,4)(2,7)(3,10)(5,9)(6,15)(8,14)(11,21)(13,22)(16,20)(17,19)(18,23)(24,25), (1,3)(2,6)(5,11)(7,9)(8,16)(10,12)(13,23)(14,22)(15,19)(17,21)(18,24)(20,25), (1,2)(3,9)(4,11)(6,7)(8,13)(10,17)(12,15)(14,23)(16,22)(18,24)(19,21)(20,25), (2,8,5,13)(3,4,10,12)(6,14,15,24)(7,18,17,23)(9,20,19,16)(11,22,21,25));