Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22×C4 |
| direct product | C31×#1969 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G | |
312 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 3 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)(36,37,38,39)(40,41)(42,47)(43,52)(44,54)(45,49)(46,50)(48,53)(51,55);(32,33,34,35)(37,39)(40,42,44,50,45,51,53,43)(41,46,48,47,49,52,54,55);(40,43,53,51,45,50,44,42)(41,47,54,46,49,55,48,52)1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#7113, C31×#13014, C31×#13015, … |
| maximal normal subgroup of | C31×#7113, C31×#13014, C31×#13015, … |
| maximal quotient of | C31×#9656, C31×#9662, C31×#9718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 701);