Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72210011 |
| commutator, hypocenter | G | C1 |
| Frattini | C26 | #11757162 |
| Fitting, cosocle | C22×C44 | A5 |
| socle | C25 | #14515301 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
7869 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 7869 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) = 72 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,5,4,10)(2,6)(7,12)(9,11)(13,28,25,21,17,14,32,20)(15,29)(19,70)(22,42,23,61,46,59,41,38)(24,40)(26,48)(31,64)(33,69,58,66,54,68,56,52)(34,50,57,71,55,65,53,62)(35,51)(36,67)(37,72)(39,49)(43,60)(44,45)(47,63);(1,5,9,4,10,12)(2,7,8,6,11,3)(13,46,31)(14,60,68,28,43,69)(15,72,63,29,67,48)(16,55,38)(17,22,45)(18,57,42)(19,62,58)(20,49,66,21,39,52)(23,26,25)(24,70,64,40,51,44)(27,53,59)(30,34,61)(32,41,47)(33,36,50)(35,71,56)(37,65,54)88 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((1,5,4,10)(2,6)(7,12)(9,11)(13,28,25,21,17,14,32,20)(15,29)(19,70)(22,42,23,61,46,59,41,38)(24,40)(26,48)(31,64)(33,69,58,66,54,68,56,52)(34,50,57,71,55,65,53,62)(35,51)(36,67)(37,72)(39,49)(43,60)(44,45)(47,63), (1,5,9,4,10,12)(2,7,8,6,11,3)(13,46,31)(14,60,68,28,43,69)(15,72,63,29,67,48)(16,55,38)(17,22,45)(18,57,42)(19,62,58)(20,49,66,21,39,52)(23,26,25)(24,70,64,40,51,44)(27,53,59)(30,34,61)(32,41,47)(33,36,50)(35,71,56)(37,65,54));