Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C35 · Schur multiplier (H2) = C310 |
| direct product | C2×#14555437 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C33×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C35 |
| prime radical | C2×C3×H(5,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32×H(3,𝔽3) ◁ G | |
12328 conjugacy classes of subgroups; only the 5698 normal subgroups are shown.
This group has 12328 conjugacy classes of subgroups — too many to draw. Only its 5698 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 — 5 generators on 83 points:
(1,2)(5,13,34)(9,25,14)(15,37,35)(16,17,42)(22,26,52)(27,36,38)(28,53,29)(39,43,40)(41,44,66)(47,56,54)(48,55,70)(57,58,60)(59,74,61)(64,67,65)(68,76,75)(69,71,82)(77,78,79)(80,83,81);(3,4,8)(5,14,22)(6,10,23)(7,12,24)(9,26,13)(11,33,21)(15,38,47)(16,29,48)(17,28,55)(18,30,49)(19,31,50)(20,32,51)(25,52,34)(27,56,37)(35,36,54)(39,60,68)(40,58,75)(41,61,69)(42,53,70)(43,57,76)(44,59,71)(45,62,72)(46,63,73)(64,79,80)(65,78,81)(66,74,82)(67,77,83);(3,5,15,7,13,35,21,34,37)(4,9,27,12,25,38,11,14,36)(6,17,43,20,42,39,19,16,40)(8,22,47,24,26,54,33,52,56)(10,29,60,32,28,58,31,53,57)(18,41,64,45,44,65,46,66,67)(23,48,68,51,55,75,50,70,76)(30,61,79,62,59,78,63,74,77)(49,71,83,72,82,80,73,69,81);(3,6,18)(4,10,30)(5,16,41)(7,20,45)(8,23,49)(9,28,59)(11,31,63)(12,32,62)(13,17,44)(14,29,61)(15,39,64)(19,46,21)(22,48,69)(24,51,72)(25,53,74)(26,55,71)(27,57,77)(33,50,73)(34,42,66)(35,40,65)(36,58,78)(37,43,67)(38,60,79)(47,68,80)(52,70,82)(54,75,81)(56,76,83);(6,19,20)(10,31,32)(16,42,17)(18,45,46)(23,50,51)(28,29,53)(30,62,63)(39,43,40)(41,44,66)(48,70,55)(49,72,73)(57,58,60)(59,74,61)(64,65,67)(68,76,75)(69,71,82)(77,79,78)(80,81,83)598 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | C2×#72221916, C2×#72221888, C2×#72221940 |
| maximal subgroup of | C2×#72221882, C2×#72221916, C2×#72221888, … |
| maximal normal subgroup of | C2×#72221882, C2×#72221916, C2×#72221888, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(5,13,34)(9,25,14)(15,37,35)(16,17,42)(22,26,52)(27,36,38)(28,53,29)(39,43,40)(41,44,66)(47,56,54)(48,55,70)(57,58,60)(59,74,61)(64,67,65)(68,76,75)(69,71,82)(77,78,79)(80,83,81), (3,4,8)(5,14,22)(6,10,23)(7,12,24)(9,26,13)(11,33,21)(15,38,47)(16,29,48)(17,28,55)(18,30,49)(19,31,50)(20,32,51)(25,52,34)(27,56,37)(35,36,54)(39,60,68)(40,58,75)(41,61,69)(42,53,70)(43,57,76)(44,59,71)(45,62,72)(46,63,73)(64,79,80)(65,78,81)(66,74,82)(67,77,83), (3,5,15,7,13,35,21,34,37)(4,9,27,12,25,38,11,14,36)(6,17,43,20,42,39,19,16,40)(8,22,47,24,26,54,33,52,56)(10,29,60,32,28,58,31,53,57)(18,41,64,45,44,65,46,66,67)(23,48,68,51,55,75,50,70,76)(30,61,79,62,59,78,63,74,77)(49,71,83,72,82,80,73,69,81), (3,6,18)(4,10,30)(5,16,41)(7,20,45)(8,23,49)(9,28,59)(11,31,63)(12,32,62)(13,17,44)(14,29,61)(15,39,64)(19,46,21)(22,48,69)(24,51,72)(25,53,74)(26,55,71)(27,57,77)(33,50,73)(34,42,66)(35,40,65)(36,58,78)(37,43,67)(38,60,79)(47,68,80)(52,70,82)(54,75,81)(56,76,83), (6,19,20)(10,31,32)(16,42,17)(18,45,46)(23,50,51)(28,29,53)(30,62,63)(39,43,40)(41,44,66)(48,70,55)(49,72,73)(57,58,60)(59,74,61)(64,65,67)(68,76,75)(69,71,82)(77,79,78)(80,81,83));