Finite Group DatabasebetaFinite Group Databasebeta| order | 58320 |
| isoclinism | isoclinic class of #72228153 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C33 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72209958 |
| commutator, hypocenter | G | C1 |
| Fitting, socle, cosocle | C2×C34 | A6 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
407 conjugacy classes of subgroups; only the 5 normal subgroups are shown.
This group has 407 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.
| minimal degree | μ(G) = 110 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 110 points:
(1,3,8)(2,6,15)(4,10,22)(5,12,23)(7,13,27)(9,20,43)(14,30,33)(16,34,52)(17,36,62)(18,39,64)(19,41,58)(21,35,60)(24,48,68)(25,51,37)(26,53,66)(28,55,76)(31,44,69)(32,57,50)(38,63,42)(45,70,79)(46,47,72)(56,75,78)(59,74,73)(61,71,65)(81,83,89)(82,86,93)(84,90,99)(85,87,97)(88,94,105)(91,102,101)(95,100,108)(96,104,98)(106,107,109);(1,9,14,8,4,2,42,22)(3,46,40,23,10,18,11,58)(5,65,73,15,19,25,78,43)(6,52,63,29,20,31,33,67)(7,55,64,30,21,26,72,38)(12,39,68,57,41,47,32,24)(13,77,75,79,35,54,74,62)(16,69,59,27,44,34,56,60)(17,70,61,28,45,36,51,66)(37,48,53,49,71,50,76,80)(81,98,107,101)(82,95,89,84)(83,94,110,96)(85,104)(86,102,106,103)(87,105,99,91)(90,108,92,97)(100,109)39 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((1,3,8)(2,6,15)(4,10,22)(5,12,23)(7,13,27)(9,20,43)(14,30,33)(16,34,52)(17,36,62)(18,39,64)(19,41,58)(21,35,60)(24,48,68)(25,51,37)(26,53,66)(28,55,76)(31,44,69)(32,57,50)(38,63,42)(45,70,79)(46,47,72)(56,75,78)(59,74,73)(61,71,65)(81,83,89)(82,86,93)(84,90,99)(85,87,97)(88,94,105)(91,102,101)(95,100,108)(96,104,98)(106,107,109), (1,9,14,8,4,2,42,22)(3,46,40,23,10,18,11,58)(5,65,73,15,19,25,78,43)(6,52,63,29,20,31,33,67)(7,55,64,30,21,26,72,38)(12,39,68,57,41,47,32,24)(13,77,75,79,35,54,74,62)(16,69,59,27,44,34,56,60)(17,70,61,28,45,36,51,66)(37,48,53,49,71,50,76,80)(81,98,107,101)(82,95,89,84)(83,94,110,96)(85,104)(86,102,106,103)(87,105,99,91)(90,108,92,97)(100,109));