Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C38 |
| direct product | C2×#14554331 |
| 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 | |
4048 conjugacy classes of subgroups; only the 622 normal subgroups are shown.
This group has 4048 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)(5,9,22)(6,10,23)(7,12,24)(11,33,21)(13,25,44)(14,26,45)(15,28,47)(16,27,46)(17,29,48)(18,30,49)(19,32,51)(20,31,50)(34,52,66)(35,54,67)(36,55,68)(37,56,69)(38,57,70)(39,53,71)(40,58,72)(41,59,73)(42,61,74)(43,60,75)(62,76,80)(63,77,81)(64,78,82)(65,79,83);(3,5,13)(4,9,25)(6,17,40)(7,15,35)(8,22,44)(10,29,58)(11,27,53)(12,28,54)(14,36,19)(16,39,21)(18,37,62)(20,38,34)(23,48,72)(24,47,67)(26,55,32)(30,56,76)(31,57,52)(33,46,71)(41,63,43)(42,64,65)(45,68,51)(49,69,80)(50,70,66)(59,77,60)(61,78,79)(73,81,75)(74,82,83);(3,6,18,7,19,42,21,20,43)(4,10,30,12,32,61,11,31,60)(5,14,37,15,38,64,16,17,41)(8,23,49,24,51,74,33,50,75)(9,26,56,28,57,78,27,29,59)(13,34,62,35,40,65,39,36,63)(22,45,69,47,70,82,46,48,73)(25,52,76,54,58,79,53,55,77)(44,66,80,67,72,83,71,68,81);(4,11,12)(5,15,16)(6,19,20)(8,24,33)(22,46,47)(23,50,51)(25,53,54)(26,29,57)(30,60,61)(44,67,71)(45,70,48)(49,74,75)(52,55,58)(56,59,78)(62,65,63)(66,72,68)(69,82,73)(80,81,83)310 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | C2×#72221912 |
| maximal subgroup of | C2×#72221849, C2×#72221912 |
| maximal normal subgroup of | C2×#72221849, C2×#72221912 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,8)(5,9,22)(6,10,23)(7,12,24)(11,33,21)(13,25,44)(14,26,45)(15,28,47)(16,27,46)(17,29,48)(18,30,49)(19,32,51)(20,31,50)(34,52,66)(35,54,67)(36,55,68)(37,56,69)(38,57,70)(39,53,71)(40,58,72)(41,59,73)(42,61,74)(43,60,75)(62,76,80)(63,77,81)(64,78,82)(65,79,83), (3,5,13)(4,9,25)(6,17,40)(7,15,35)(8,22,44)(10,29,58)(11,27,53)(12,28,54)(14,36,19)(16,39,21)(18,37,62)(20,38,34)(23,48,72)(24,47,67)(26,55,32)(30,56,76)(31,57,52)(33,46,71)(41,63,43)(42,64,65)(45,68,51)(49,69,80)(50,70,66)(59,77,60)(61,78,79)(73,81,75)(74,82,83), (3,6,18,7,19,42,21,20,43)(4,10,30,12,32,61,11,31,60)(5,14,37,15,38,64,16,17,41)(8,23,49,24,51,74,33,50,75)(9,26,56,28,57,78,27,29,59)(13,34,62,35,40,65,39,36,63)(22,45,69,47,70,82,46,48,73)(25,52,76,54,58,79,53,55,77)(44,66,80,67,72,83,71,68,81), (4,11,12)(5,15,16)(6,19,20)(8,24,33)(22,46,47)(23,50,51)(25,53,54)(26,29,57)(30,60,61)(44,67,71)(45,70,48)(49,74,75)(52,55,58)(56,59,78)(62,65,63)(66,72,68)(69,82,73)(80,81,83));