Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C2×C4 |
| direct product | C29×#1833 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #117 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C2×C29 | #382 |
| prime radical | C23×C29 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C29×#117 ◁ G | |
172 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 61 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)(32,40,43,36)(33,44,46,59)(34,39)(35,38)(37,45,50,60)(41,54)(42,53,57,49)(47,51)(48,55)(52,61)(56,58);(30,32,34,42,35,43,47,57)(31,36,39,49,38,40,51,53)(33,41,45,55,46,56,60,61)(37,48,44,58,50,52,59,54);(30,33)(31,37,38,50)(32,41)(34,45)(35,46)(36,48,40,52)(39,44,51,59)(42,55)(43,56)(47,60)(49,58,53,54)(57,61)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12826, C29×#12827, C29×#12828, … |
| maximal normal subgroup of | C29×#12826, C29×#12827, C29×#12828, … |
| maximal quotient of | C29×#8780, C29×#8781, C29×#8782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 565);