Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C26 |
| direct product | C31×#3004 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C31×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G | |
738 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 224 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 — 4 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)(38,44)(39,47)(40,42)(41,45)(43,49)(46,51)(48,50);(36,38,40,46)(37,41,42,50)(39,48,49,45)(43,51,47,44);(33,35);(37,42)(41,50)(43,47)(44,51)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#9519, C31×#18136, C31×#18146, … |
| maximal normal subgroup of | C31×#9519, C31×#18136, C31×#18146, … |
| maximal quotient of | C31×#18133, C31×#18138, C31×#18143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1736);