Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C35 |
| direct product | C2×#72221895 |
| Fitting | C2×#14554344 ⋊ C2 |
| derived | C34 ⋊ C22×C33 |
| Sylow | #14554344 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72221895 |
| commutator, cosocle, hypocenter | C34 | C22×C33 |
| Frattini | C33 | C2×C33×D3 |
| Fitting | C2×#14554344 | C2 |
| socle | C2×C3 | C3×#92258 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C2 | |
3314 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 3314 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 83 points:
(2,20,32,6,9,55)(3,39,12,11,13,38)(4,46,48,15,16,47)(5,17)(7,50,65,18,26,33)(8,53,34,19,29,59)(10,72,25,35,36,49)(14,44)(21,64,52,30,58,68)(22,63,56,37,60,31)(23,62,71,24,57,81)(27,80,42,66,67,54)(28,73,70,51,69,61)(40,77,75,45,74,79)(41,76)(43,78);(1,2,21,5,7,60,17,24,51)(3,25,49,12,42,19,13,8,54)(4,10,59,16,27,80,48,29,72)(6,11,63,23,39,30,18,38,28)(9,52,74,26,22,40,62,61,75)(14,64,71,41,69,33,43,31,55)(15,37,20,47,68,57,46,70,50)(32,56,79,65,73,45,81,58,77)(34,66,76,36,35,44,67,53,78)(82,83);(2,7,24)(6,18,23)(8,25,42)(9,26,62)(10,27,29)(19,49,54)(20,50,57)(21,51,60)(22,52,61)(28,63,30)(31,64,69)(32,65,81)(33,71,55)(34,36,67)(35,66,53)(37,68,70)(56,58,73)(59,72,80)236 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72221898 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,20,32,6,9,55)(3,39,12,11,13,38)(4,46,48,15,16,47)(5,17)(7,50,65,18,26,33)(8,53,34,19,29,59)(10,72,25,35,36,49)(14,44)(21,64,52,30,58,68)(22,63,56,37,60,31)(23,62,71,24,57,81)(27,80,42,66,67,54)(28,73,70,51,69,61)(40,77,75,45,74,79)(41,76)(43,78), (1,2,21,5,7,60,17,24,51)(3,25,49,12,42,19,13,8,54)(4,10,59,16,27,80,48,29,72)(6,11,63,23,39,30,18,38,28)(9,52,74,26,22,40,62,61,75)(14,64,71,41,69,33,43,31,55)(15,37,20,47,68,57,46,70,50)(32,56,79,65,73,45,81,58,77)(34,66,76,36,35,44,67,53,78)(82,83), (2,7,24)(6,18,23)(8,25,42)(9,26,62)(10,27,29)(19,49,54)(20,50,57)(21,51,60)(22,52,61)(28,63,30)(31,64,69)(32,65,81)(33,71,55)(34,36,67)(35,66,53)(37,68,70)(56,58,73)(59,72,80));