Finite Group DatabasebetaFinite Group Databasebeta| order | 3776 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C59 · Schur multiplier (H2) = C25 |
| direct product | C59×#575 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C59 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C59 |
| socle | C2×C59 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C59 ◁ C22×C4×C59 ◁ G | |
262 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 944 |
A minimal faithful permutation representation — 4 generators on 75 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)(62,69)(63,68)(64,72)(65,67)(66,70)(71,75)(73,74);(60,62)(61,66)(63,70)(64,73)(65,71)(67,74)(68,75)(69,72);(61,67)(66,74)(68,72)(69,75);(60,63,65,64)(61,68,67,72)(62,70,71,73)(66,75,74,69)1298 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C59×#1881, C59×#1906, C59×#2008, … |
| maximal normal subgroup of | C59×#1881, C59×#1906, C59×#2008, … |
| maximal quotient of | C59×#2896, C59×#2911, C59×#2943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3776, 256);