Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C2×C8 |
| direct product | C29×#1721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C29 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C42×C29 ◁ G | |
242 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 45 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)(31,34)(32,36)(35,37)(39,42)(40,43)(44,45);(30,31,32,34,33,35,36,37);(31,35)(34,37)(38,39,43,45,41,44,40,42)754 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#7527, C29×#12314, C2×C29×#1721, … |
| maximal normal subgroup of | C29×#7527, C29×#12314, C2×C29×#1721, … |
| maximal quotient of | C29×#8820, C29×#9830, C29×#9832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 453);