Finite Group DatabasebetaFinite Group Databasebeta| order | 3776 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C59 · Schur multiplier (H2) = C4 |
| direct product | C59×#482 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C59 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C59 |
| prime radical | #121×C59 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C22×C4×C59 ◁ G | |
106 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 106 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) = 1888 |
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)(60,61)(62,65)(63,64)(66,67)(69,72)(70,73)(74,75);(61,64)(65,67)(68,69,71,72)(70,73,74,75);(60,62,63,66)(61,65,64,67)(68,70,71,74)(69,73,72,75)1121 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C59×#482, C59×#3091, C59×#3093, … |
| maximal normal subgroup of | C2×C59×#482, C59×#3091, C59×#3093, … |
| maximal quotient of | C59×#1825, C59×#1854, C59×#1875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3776, 163);