Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C38 |
| direct product | C2×#14554330 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C3×C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C2×C34 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C34 ◁ G | |
4444 conjugacy classes of subgroups; only the 622 normal subgroups are shown.
This group has 4444 conjugacy classes of subgroups — too many to draw. Only its 622 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 — 4 generators on 83 points:
(1,2)(3,4,8)(5,9,21)(6,10,22)(7,11,24)(12,25,44)(13,26,45)(14,28,46)(15,27,47)(16,29,48)(17,30,49)(18,43,51)(19,31,50)(20,32,23)(33,52,66)(34,53,68)(35,54,69)(36,55,70)(37,56,71)(38,57,67)(39,58,72)(40,59,73)(41,61,74)(42,60,75)(62,76,80)(63,77,81)(64,78,82)(65,79,83);(3,5,12)(4,9,25)(6,16,33)(7,15,34)(8,21,44)(10,29,52)(11,27,53)(13,35,19)(14,38,20)(17,40,62)(18,37,63)(22,48,66)(23,46,67)(24,47,68)(26,54,31)(28,57,32)(30,59,76)(36,65,41)(39,64,42)(43,56,77)(45,69,50)(49,73,80)(51,71,81)(55,79,61)(58,78,60)(70,83,74)(72,82,75);(3,6,17,20,18,41,7,19,42)(4,10,30,32,43,61,11,31,60)(5,13,36,14,16,39,15,37,40)(8,22,49,23,51,74,24,50,75)(9,26,55,28,29,58,27,56,59)(12,33,62,38,63,65,34,35,64)(21,45,70,46,48,72,47,71,73)(25,52,76,57,77,79,53,54,78)(44,66,80,67,81,83,68,69,82);(4,11,32)(6,18,19)(8,23,24)(9,27,28)(13,16,37)(21,46,47)(22,50,51)(25,53,57)(30,60,61)(33,63,35)(44,67,68)(45,71,48)(49,74,75)(55,59,58)(66,69,81)(70,72,73)(76,78,79)(80,83,82)310 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | C2×#72221919 |
| maximal subgroup of | C2×#72221849, C2×#72221919 |
| maximal normal subgroup of | C2×#72221849, C2×#72221919 |
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)(6,10,22)(7,11,24)(12,25,44)(13,26,45)(14,28,46)(15,27,47)(16,29,48)(17,30,49)(18,43,51)(19,31,50)(20,32,23)(33,52,66)(34,53,68)(35,54,69)(36,55,70)(37,56,71)(38,57,67)(39,58,72)(40,59,73)(41,61,74)(42,60,75)(62,76,80)(63,77,81)(64,78,82)(65,79,83), (3,5,12)(4,9,25)(6,16,33)(7,15,34)(8,21,44)(10,29,52)(11,27,53)(13,35,19)(14,38,20)(17,40,62)(18,37,63)(22,48,66)(23,46,67)(24,47,68)(26,54,31)(28,57,32)(30,59,76)(36,65,41)(39,64,42)(43,56,77)(45,69,50)(49,73,80)(51,71,81)(55,79,61)(58,78,60)(70,83,74)(72,82,75), (3,6,17,20,18,41,7,19,42)(4,10,30,32,43,61,11,31,60)(5,13,36,14,16,39,15,37,40)(8,22,49,23,51,74,24,50,75)(9,26,55,28,29,58,27,56,59)(12,33,62,38,63,65,34,35,64)(21,45,70,46,48,72,47,71,73)(25,52,76,57,77,79,53,54,78)(44,66,80,67,81,83,68,69,82), (4,11,32)(6,18,19)(8,23,24)(9,27,28)(13,16,37)(21,46,47)(22,50,51)(25,53,57)(30,60,61)(33,63,35)(44,67,68)(45,71,48)(49,74,75)(55,59,58)(66,69,81)(70,72,73)(76,78,79)(80,83,82));