Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C25 |
| direct product | C2×C31×#480 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C31 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C2×C31×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G | |
512 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,17)(16,18);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(12,15)(13,18)(16,17)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#9043, C31×#9260, C31×#9918, … |
| maximal normal subgroup of | C31×#9043, C31×#9260, C31×#9918, … |
| maximal quotient of | C2×C31×#1823, C2×C31×#1852, C2×C31×#1874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1817);