Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2191 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C31 |
| prime radical | C31×#575 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×M16×C31 ◁ G | |
216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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)(32,33,37,41)(34,42,44,60)(35,46,48,62)(36,51,49,40)(38,53,54,43)(39,56,58,50)(45,61,52,57)(47,59,63,55);(32,34)(33,38)(35,47)(36,52)(37,44)(39,57)(40,59)(41,54)(42,46)(43,50)(45,49)(48,63)(51,55)(53,56)(58,61)(60,62);(35,48)(39,58)(42,60)(43,53)(46,62)(47,63)(50,56)(57,61)620 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12115, C31×#12116, C31×#12118, … |
| maximal normal subgroup of | C31×#12115, C31×#12116, C31×#12118, … |
| maximal quotient of | C31×#12105, C31×#12106, C31×#12107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 923);