Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C35 · Schur multiplier (H2) = C310 |
| direct product | C2×#14555436 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C33×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C35 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32×H(3,𝔽3) ◁ G | |
12724 conjugacy classes of subgroups; only the 5698 normal subgroups are shown.
This group has 12724 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)(3,4,8)(5,9,21)(6,11,23)(7,12,22)(10,28,20)(13,24,40)(14,27,43)(15,25,42)(16,26,41)(17,31,44)(18,30,46)(19,29,45)(32,48,61)(33,49,60)(34,52,64)(35,51,65)(36,50,63)(37,47,62)(38,54,66)(39,53,67)(55,69,76)(56,68,77)(57,70,75)(58,72,79)(59,71,78)(73,81,82)(74,80,83);(3,5,13)(4,9,24)(6,15,32)(7,14,33)(8,21,40)(10,26,47)(11,25,48)(12,27,49)(16,37,20)(17,34,57)(18,35,55)(19,36,56)(22,43,60)(23,42,61)(28,41,62)(29,50,68)(30,51,69)(31,52,70)(38,58,73)(39,59,74)(44,64,75)(45,63,77)(46,65,76)(53,71,80)(54,72,81)(66,79,82)(67,78,83);(4,10,12)(5,14,16)(6,17,19)(8,22,28)(21,41,43)(23,45,44)(24,47,49)(25,50,52)(30,53,54)(40,60,62)(42,64,63)(46,66,67)(48,68,70)(51,71,72)(55,73,74)(61,75,77)(65,79,78)(76,83,82);(3,6,18)(4,11,30)(5,15,35)(7,17,38)(8,23,46)(9,25,51)(10,29,53)(12,31,54)(13,32,55)(14,34,58)(16,36,59)(19,39,20)(21,42,65)(22,44,66)(24,48,69)(26,50,71)(27,52,72)(28,45,67)(33,57,73)(37,56,74)(40,61,76)(41,63,78)(43,64,79)(47,68,80)(49,70,81)(60,75,82)(62,77,83);(6,19,17)(11,29,31)(15,36,34)(18,38,39)(23,45,44)(25,50,52)(30,54,53)(32,56,57)(35,58,59)(42,63,64)(46,66,67)(48,68,70)(51,72,71)(55,73,74)(61,77,75)(65,79,78)(69,81,80)(76,82,83)598 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | C2×#72221892, C2×#72221874, C2×#72221905 |
| maximal subgroup of | C2×#72221849, C2×#72221892, C2×#72221874, … |
| maximal normal subgroup of | C2×#72221849, C2×#72221892, C2×#72221874, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,8)(5,9,21)(6,11,23)(7,12,22)(10,28,20)(13,24,40)(14,27,43)(15,25,42)(16,26,41)(17,31,44)(18,30,46)(19,29,45)(32,48,61)(33,49,60)(34,52,64)(35,51,65)(36,50,63)(37,47,62)(38,54,66)(39,53,67)(55,69,76)(56,68,77)(57,70,75)(58,72,79)(59,71,78)(73,81,82)(74,80,83), (3,5,13)(4,9,24)(6,15,32)(7,14,33)(8,21,40)(10,26,47)(11,25,48)(12,27,49)(16,37,20)(17,34,57)(18,35,55)(19,36,56)(22,43,60)(23,42,61)(28,41,62)(29,50,68)(30,51,69)(31,52,70)(38,58,73)(39,59,74)(44,64,75)(45,63,77)(46,65,76)(53,71,80)(54,72,81)(66,79,82)(67,78,83), (4,10,12)(5,14,16)(6,17,19)(8,22,28)(21,41,43)(23,45,44)(24,47,49)(25,50,52)(30,53,54)(40,60,62)(42,64,63)(46,66,67)(48,68,70)(51,71,72)(55,73,74)(61,75,77)(65,79,78)(76,83,82), (3,6,18)(4,11,30)(5,15,35)(7,17,38)(8,23,46)(9,25,51)(10,29,53)(12,31,54)(13,32,55)(14,34,58)(16,36,59)(19,39,20)(21,42,65)(22,44,66)(24,48,69)(26,50,71)(27,52,72)(28,45,67)(33,57,73)(37,56,74)(40,61,76)(41,63,78)(43,64,79)(47,68,80)(49,70,81)(60,75,82)(62,77,83), (6,19,17)(11,29,31)(15,36,34)(18,38,39)(23,45,44)(25,50,52)(30,54,53)(32,56,57)(35,58,59)(42,63,64)(46,66,67)(48,68,70)(51,72,71)(55,73,74)(61,77,75)(65,79,78)(69,81,80)(76,82,83));