Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2045 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C22×D8×C31 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C22×C4×C31 ◁ G | |
296 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 84 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)(38,45)(39,41)(40,44)(42,46)(43,48)(47,50)(49,51);(37,41)(38,46)(39,48)(42,50)(43,44)(45,49);(33,35)(36,38,39,47,40,46,48,51)(37,42,43,50,44,45,41,49)806 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12164, C31×#12165, C31×#12166, … |
| maximal normal subgroup of | C31×#12164, C31×#12165, C31×#12166, … |
| maximal quotient of | C31×#10440, C31×#10441, C31×#10448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 777);