Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#2980 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C24 |
| commutator | C22 | C23×C4×C31 |
| Frattini, cosocle | C2×C4 | C24×C31 |
| socle | C22×C31 | C23×C4 |
| prime radical | C31×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ G | |
394 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 1984 |
A minimal faithful permutation representation — 4 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)(32,33)(34,38)(35,37)(36,39)(40,41,43,47,44,48,51,55)(42,45,50,52,49,46,53,54);(32,34)(33,36)(35,39)(37,38)(40,42)(41,45)(43,50)(44,49)(46,48)(47,52)(51,53)(54,55);(34,39)(36,38)(40,42)(41,46)(43,50)(44,49)(45,48)(47,54)(51,53)(52,55);(33,37)(36,38)(42,49)(45,46)(50,53)(52,54)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#8918, C31×#9005, C31×#9007, … |
| maximal normal subgroup of | C31×#8918, C31×#9005, C31×#9007, … |
| maximal quotient of | C31×#20532, C31×#20539, C31×#20563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1712);