Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C2×C42 |
| direct product | C29×#2835 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| prime radical | C24×C29 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
380 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 4 generators on 53 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)(35,37)(38,39)(40,42)(41,43)(44,45)(46,47,50,52)(48,49,51,53);(32,36)(35,37)(38,39,41,43)(40,42,45,44)(46,48)(47,49)(50,51)(52,53);(30,31,33,34)(32,35,36,37)(38,40,41,45)(39,42,43,44)(46,49)(47,51)(48,52)(50,53);(30,32)(31,35)(33,36)(34,37)(38,39)(40,44)(41,43)(42,45)754 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12139, C29×#12233, C2×C29×#2835, … |
| maximal normal subgroup of | C29×#12139, C29×#12233, C2×C29×#2835, … |
| maximal quotient of | C29×#14627, C29×#14635, C29×#14647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1567);