Finite Group DatabasebetaFinite Group Databasebeta| order | 3776 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C59 · Schur multiplier (H2) = C22 |
| direct product | C59×#471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C59 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C59 |
| prime radical | AGL(1,ℤ/8)×C59 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C59 ◁ C2×C4×C59 ◁ G | |
104 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 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)(61,65)(62,68)(63,71)(67,73)(69,70)(72,74);(60,61,63,65,64,67,71,73)(62,66,69,72,70,75,68,74);(60,62)(61,66)(63,69)(64,70)(65,72)(67,75)(68,71)(73,74)944 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C59×#471, C59×#3067, C59×#3068, … |
| maximal normal subgroup of | C2×C59×#471, C59×#3067, C59×#3068, … |
| maximal quotient of | C59×#1665, C59×#1666, C59×#1671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3776, 152);