Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2112 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C31 | C23 |
| commutator | C22 | C22×C8×C31 |
| Frattini, cosocle | C2×C8 | C23×C31 |
| socle | C22×C31 | C22×C8 |
| prime radical | C24×C31 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C31 ◁ G | |
260 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 128 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)(32,33)(34,35)(36,37,38,42,39,43,44,49,40,41,45,47,46,48,50,51);(33,35);(37,41)(42,47)(43,48)(49,51)1736 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#7184, C31×#7185, C31×#7200, … |
| maximal normal subgroup of | C31×#7184, C31×#7185, C31×#7200, … |
| maximal quotient of | C31×#11418, C31×#11440, C31×#11460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 844);