Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22×C4 |
| direct product | C31×#1812 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #117 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C2×C31 | #380 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C2×M16×C31 ◁ G | |
260 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 63 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)(34,42)(35,46,48,39)(36,49)(37,41)(38,44)(40,50)(43,56,58,54)(45,55)(47,53,62,61)(51,57,60,63)(52,59);(32,34,36,44,37,45,50,59)(33,38,40,42,41,52,49,55)(35,43,47,57,48,58,62,63)(39,51,53,54,46,60,61,56);(32,35)(33,39)(34,43)(36,47)(37,48)(38,51)(40,53)(41,46)(42,54)(44,57)(45,58)(49,61)(50,62)(52,60)(55,56)(59,63)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7227, C31×#7228, C31×#11867, … |
| maximal normal subgroup of | C31×#7227, C31×#7228, C31×#11867, … |
| maximal quotient of | C31×#8601, C31×#8602, C31×#8603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 544);