Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#3363 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C31 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C4×Q8×C31 ◁ G | |
336 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 192 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)(33,36)(37,38)(40,41,44,47)(42,48,50,55)(43,51,53,46)(45,52,54,49);(32,33,35,36)(34,38,39,37);(40,42,44,50)(41,45,47,54)(43,52,53,49)(46,55,51,48);(32,34,35,39)(33,37,36,38)1085 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#19812, C31×#19816, C31×#19817, … |
| maximal normal subgroup of | C31×#19812, C31×#19816, C31×#19817, … |
| maximal quotient of | Q8×C31×#103, C31×#22542, C31×#22740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 2095);