Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| isoclinism | isoclinic class of #72238105 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini | C26 | #11757163 |
| Fitting, cosocle | C2×#158995 | A5 |
| socle | C24 | #72209606 |
| LCS | G | |
| UCS | C1 | |
3615 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 3615 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 80 points:
(1,66,25,22,2,41)(3,40,60,8,7,9)(4,27,29,6,47,36)(5,17,23,51,10,38)(11,46,26,33,15,30)(12,43,21,34,16,37)(13,44)(14,42,65,55,20,53)(18,67,56,45,48,19)(24,70,77,39,64,57)(28,31,79,50,52,71)(32,54,35,49,73,63)(58,78,62,61,76,59)(68,69,80,75,72,74);(1,46,21,3)(2,42,30,6,40,18,34,10)(4,26,20,66,17,12,45,7)(5,35,19,54,36,44,14,49)(8,22,15,37)(9,28,60,68,11,24,33,62)(13,29,73,56,63,51,32,55)(16,50,43,59,25,39,41,75)(23,72,47,58)(27,61,38,69)(31,67,52,65)(48,64,53,70)(71,79)(74,78,76,80)50 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,66,25,22,2,41)(3,40,60,8,7,9)(4,27,29,6,47,36)(5,17,23,51,10,38)(11,46,26,33,15,30)(12,43,21,34,16,37)(13,44)(14,42,65,55,20,53)(18,67,56,45,48,19)(24,70,77,39,64,57)(28,31,79,50,52,71)(32,54,35,49,73,63)(58,78,62,61,76,59)(68,69,80,75,72,74), (1,46,21,3)(2,42,30,6,40,18,34,10)(4,26,20,66,17,12,45,7)(5,35,19,54,36,44,14,49)(8,22,15,37)(9,28,60,68,11,24,33,62)(13,29,73,56,63,51,32,55)(16,50,43,59,25,39,41,75)(23,72,47,58)(27,61,38,69)(31,67,52,65)(48,64,53,70)(71,79)(74,78,76,80));