Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C32 |
| direct product | C2×#14550737 |
| 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)(3,4,9,7,12,33,28,27,40,8,14,34,29,25,59,66,62,11,30,13,23,32,6,21,57,24,41)(5,15,43,18,10,35,54,22,50,20,45,75,26,38,37,64,17,31,19,51,58,55,42,39,65,52,61)(16,46,73,48,77,81,80,72,74,49,60,70,53,44,67,82,71,76,56,47,68,63,78,79,83,36,69);(3,5,16,29,55,48,57,64,80,8,20,49,32,18,53,28,65,82,30,19,56,7,26,63,66,54,83)(4,10,36,6,22,60,62,45,78,14,38,72,12,17,47,24,51,77,13,42,71,25,52,46,27,15,44)(9,31,67,33,43,74,40,37,73,34,61,79,59,75,76,11,39,70,23,50,81,21,58,69,41,35,68);(3,6,23,29,62,21,57,14,41,8,12,9,32,24,33,28,13,40,30,25,34,7,27,59,66,4,11)(5,17,50,55,51,58,64,42,35,20,52,31,18,15,43,65,10,37,19,22,61,26,45,75,54,38,39)(16,47,67,48,77,74,80,71,73,49,46,79,53,44,76,82,36,70,56,60,81,63,78,69,83,72,68)214 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72221818 |
| maximal normal subgroup of | C2×#72221818 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,9,7,12,33,28,27,40,8,14,34,29,25,59,66,62,11,30,13,23,32,6,21,57,24,41)(5,15,43,18,10,35,54,22,50,20,45,75,26,38,37,64,17,31,19,51,58,55,42,39,65,52,61)(16,46,73,48,77,81,80,72,74,49,60,70,53,44,67,82,71,76,56,47,68,63,78,79,83,36,69), (3,5,16,29,55,48,57,64,80,8,20,49,32,18,53,28,65,82,30,19,56,7,26,63,66,54,83)(4,10,36,6,22,60,62,45,78,14,38,72,12,17,47,24,51,77,13,42,71,25,52,46,27,15,44)(9,31,67,33,43,74,40,37,73,34,61,79,59,75,76,11,39,70,23,50,81,21,58,69,41,35,68), (3,6,23,29,62,21,57,14,41,8,12,9,32,24,33,28,13,40,30,25,34,7,27,59,66,4,11)(5,17,50,55,51,58,64,42,35,20,52,31,18,15,43,65,10,37,19,22,61,26,45,75,54,38,39)(16,47,67,48,77,74,80,71,73,49,46,79,53,44,76,82,36,70,56,60,81,63,78,69,83,72,68));