Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C32 |
| direct product | C2×#14550739 |
| 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)(3,4,6,7,10,20,25,19,54,8,12,22,26,34,58,64,57,63,27,11,21,59,33,24,55,56,62)(5,13,36,16,9,28,51,30,47,17,39,50,23,31,67,61,32,15,18,45,46,52,35,66,38,72,48)(14,40,77,42,78,41,79,69,74,43,70,83,49,37,44,75,73,81,53,71,65,60,76,80,82,29,68);(3,5,14,26,52,42,55,61,79,8,17,43,59,16,49,25,38,75,27,18,53,7,23,60,64,51,82)(4,9,29,33,30,70,57,39,76,12,31,69,10,32,71,56,45,78,11,35,73,34,72,40,19,13,37)(6,15,44,20,36,74,54,67,77,22,48,80,58,50,81,63,66,83,21,47,41,24,46,68,62,28,65);(3,6,19,55,54,33,59,58,12,27,21,56,64,62,34,26,20,4,8,22,57,25,63,10,7,24,11)(5,15,45,61,67,72,16,50,9,18,47,39,51,28,32,52,36,35,17,48,13,38,66,30,23,46,31)(14,41,78,79,65,40,49,74,29,53,80,76,82,83,71,42,68,73,43,44,37,75,77,70,60,81,69)214 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#72221842 |
| maximal subgroup of | C2×#72221818, C2×#72221842 |
| maximal normal subgroup of | C2×#72221818, C2×#72221842 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,6,7,10,20,25,19,54,8,12,22,26,34,58,64,57,63,27,11,21,59,33,24,55,56,62)(5,13,36,16,9,28,51,30,47,17,39,50,23,31,67,61,32,15,18,45,46,52,35,66,38,72,48)(14,40,77,42,78,41,79,69,74,43,70,83,49,37,44,75,73,81,53,71,65,60,76,80,82,29,68), (3,5,14,26,52,42,55,61,79,8,17,43,59,16,49,25,38,75,27,18,53,7,23,60,64,51,82)(4,9,29,33,30,70,57,39,76,12,31,69,10,32,71,56,45,78,11,35,73,34,72,40,19,13,37)(6,15,44,20,36,74,54,67,77,22,48,80,58,50,81,63,66,83,21,47,41,24,46,68,62,28,65), (3,6,19,55,54,33,59,58,12,27,21,56,64,62,34,26,20,4,8,22,57,25,63,10,7,24,11)(5,15,45,61,67,72,16,50,9,18,47,39,51,28,32,52,36,35,17,48,13,38,66,30,23,46,31)(14,41,78,79,65,40,49,74,29,53,80,76,82,83,71,42,68,73,43,44,37,75,77,70,60,81,69));