Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#2221 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C29 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C2×C4×C29 ◁ C29×#131 ◁ G | |
148 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 60 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)(31,37)(32,43)(33,38)(34,35)(39,54)(40,51)(41,50)(42,52)(44,58)(45,48)(46,47)(49,55)(56,61)(57,60);(30,31,38,51,35,41,47,52,36,42,46,50,34,40,33,37)(32,39,53,54,43,55,58,60,45,56,59,61,48,57,44,49);(30,32)(31,39)(33,44)(34,48)(35,43)(36,45)(37,49)(38,53)(40,57)(41,55)(42,56)(46,59)(47,58)(50,61)(51,54)(52,60)754 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#13669, C29×#13670, C29×#13700, … |
| maximal normal subgroup of | C29×#13669, C29×#13670, C29×#13700, … |
| maximal quotient of | C29×#12430, C29×#12433, C29×#12439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 953);