Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C4 |
| direct product | C31×#1694 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C31 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C31 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C42×C31 ◁ G | |
152 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 3968 |
A minimal faithful permutation representation — 3 generators on 55 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)(33,36)(34,37)(38,39)(40,41,44,47)(42,48,50,55)(43,51,53,46)(45,52,54,49);(32,33,37,39,35,38,34,36);(33,38)(36,39)(40,42,44,50)(41,45,47,54)(43,52,53,49)(46,55,51,48)713 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12390, C31×#12391, C31×#12414, … |
| maximal normal subgroup of | C31×#12390, C31×#12391, C31×#12414, … |
| maximal quotient of | C31×#8653, C31×#9226, C31×#9574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 426);