Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C32 |
| direct product | C2×#14550736 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #10624565 |
| commutator, Frattini, cosocle | C32×C9 | C2×C33 |
| prime radical | C2×C3×H(3,𝔽3) | 3−1+2 |
| LCS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32×C9 ◁ G | |
516 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 154 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:
(1,2)(5,13,33,16,37,36,17,38,66)(6,18,46)(9,25,55,14,39,57,28,34,67)(10,29,19)(15,41,69,44,65,70,45,74,35)(20,49,43)(22,51,75,53,62,77,26,58,80)(23,47,30)(27,60,68,42,73,71,40,72,56)(31,48,50)(52,78,79,59,81,83,61,82,76)(54,63,64);(3,4,8)(5,14,22)(6,19,47)(7,12,24)(9,26,17)(10,30,18)(11,32,21)(13,34,58)(15,42,52)(16,28,53)(20,50,63)(23,46,29)(25,51,37)(27,61,45)(31,64,49)(33,55,77)(35,56,79)(36,57,80)(38,39,62)(40,59,44)(41,72,82)(43,48,54)(60,78,65)(66,67,75)(68,83,69)(70,71,76)(73,81,74);(3,5,15,43,33,65,18,38,69,7,17,44,49,66,74,46,37,70,21,16,45,20,36,41,6,13,35)(4,9,27,48,67,72,10,25,56,12,28,42,31,57,60,29,34,68,11,14,40,50,55,73,19,39,71)(8,22,52,54,77,78,30,62,83,24,26,59,64,75,81,23,51,76,32,53,61,63,80,82,47,58,79)214 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#72221843 |
| maximal subgroup of | C2×#72221818, C2×#72221843 |
| maximal normal subgroup of | C2×#72221818, C2×#72221843 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(5,13,33,16,37,36,17,38,66)(6,18,46)(9,25,55,14,39,57,28,34,67)(10,29,19)(15,41,69,44,65,70,45,74,35)(20,49,43)(22,51,75,53,62,77,26,58,80)(23,47,30)(27,60,68,42,73,71,40,72,56)(31,48,50)(52,78,79,59,81,83,61,82,76)(54,63,64), (3,4,8)(5,14,22)(6,19,47)(7,12,24)(9,26,17)(10,30,18)(11,32,21)(13,34,58)(15,42,52)(16,28,53)(20,50,63)(23,46,29)(25,51,37)(27,61,45)(31,64,49)(33,55,77)(35,56,79)(36,57,80)(38,39,62)(40,59,44)(41,72,82)(43,48,54)(60,78,65)(66,67,75)(68,83,69)(70,71,76)(73,81,74), (3,5,15,43,33,65,18,38,69,7,17,44,49,66,74,46,37,70,21,16,45,20,36,41,6,13,35)(4,9,27,48,67,72,10,25,56,12,28,42,31,57,60,29,34,68,11,14,40,50,55,73,19,39,71)(8,22,52,54,77,78,30,62,83,24,26,59,64,75,81,23,51,76,32,53,61,63,80,82,47,58,79));