Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C25 |
| direct product | C31×#2327 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C31 | C24 |
| commutator | C22 | C23×C4×C31 |
| Frattini, cosocle | C23 | C24×C31 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ G | |
516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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 — 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)(33,35)(36,37,39,38)(41,45)(42,48)(43,51)(53,55);(36,38,39,37)(40,41)(42,49)(43,47)(44,45)(46,48)(50,55)(51,52)(53,54);(32,33,34,35)(40,42)(41,46)(43,50)(44,48)(45,49)(47,53)(51,54)(52,55);(40,43,44,51)(41,47,45,52)(42,50,48,54)(46,53,49,55)1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#9100, C31×#9104, C31×#9296, … |
| maximal normal subgroup of | C31×#9100, C31×#9104, C31×#9296, … |
| maximal quotient of | C31×#13779, C31×#13786, C31×#13790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1059);