Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C31 · Schur multiplier (H2) = C23×C4 |
| direct product | C31×#1740 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C22⋊C4 |
| commutator | C22 | C2×C42×C31 |
| Frattini, cosocle | C24 | C23×C31 |
| socle | C22×C31 | C2×C22⋊C4 |
| prime radical | C24×C31 | C23 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C23×C4×C31 ◁ G | |
420 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(36,37)(38,44,46,41)(39,42)(40,43)(45,50,51,48)(47,49);(32,33,34,35)(36,38)(37,41)(39,45)(40,46)(42,48)(43,44)(47,51)(49,50);(36,39,40,47)(37,42,43,49)(38,45,46,51)(41,48,44,50)1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#7252, C31×#10711, C31×#10772, … |
| maximal normal subgroup of | C31×#7252, C31×#10711, C31×#10772, … |
| maximal quotient of | C31×#7906, C31×#7908, C31×#7910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 472);