Finite Group DatabasebetaFinite Group Databasebeta| order | 3776 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C59 · Schur multiplier (H2) = C22 |
| direct product | C59×#440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C59 | C2×D4 |
| commutator | C4 | C22×C4×C59 |
| Frattini, cosocle | C2×C4 | C23×C59 |
| prime radical | C2×D4×C59 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C42×C59 ◁ G | |
120 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 1888 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59)(60,61,63,62)(64,65)(66,68)(67,69)(70,71);(65,68)(66,71)(69,70);(60,62,63,61)(65,69)(68,70)1298 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C59×#440, C59×#2943, C4×AGL(1,ℤ/8)×C59, … |
| maximal normal subgroup of | C2×C59×#440, C59×#2943, C4×AGL(1,ℤ/8)×C59, … |
| maximal quotient of | C59×SD16⋊C8, C59×#1585, C59×#1588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3776, 121);