Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2115 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C31 | C22⋊C4 |
| commutator | C22 | C22×C8×C31 |
| Frattini, cosocle | C2×C8 | C23×C31 |
| socle | C2×C31 | C2×C22⋊C8 |
| prime radical | C2×#41×C31 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C31 ◁ C22×C8×C31 ◁ G | |
220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 220 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)(32,33,35,40,36,41,47,55,37,39,48,52,49,53,60,62)(34,42,44,54,45,56,58,63,43,38,46,50,57,51,59,61);(32,34)(33,38)(35,46)(36,45)(37,43)(39,42)(40,54)(41,51)(44,48)(47,59)(49,57)(50,52)(53,56)(55,63)(58,60)(61,62);(34,43)(38,42)(44,46)(45,57)(50,54)(51,56)(58,59)(61,63)1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#7207, C31×#7208, C31×#11483, … |
| maximal normal subgroup of | C31×#7207, C31×#7208, C31×#11483, … |
| maximal quotient of | C31×#11419, C31×#11441, C31×#11461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 847);