Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C2 |
| direct product | C29×#2089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C23×C29 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ G | |
176 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 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)(30,31)(32,37)(33,35)(34,36)(38,39,41,46,43,47,42,45)(40,44,49,52,50,51,48,53);(30,32,34,33)(31,35,36,37)(38,40)(39,44)(41,49)(42,48)(43,50)(45,53)(46,52)(47,51);(30,33,34,32)(31,35,36,37)(39,45)(40,48)(41,42)(44,51)(46,47)(49,50)754 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12294, C29×#12295, C29×#12296, … |
| maximal normal subgroup of | C29×#12294, C29×#12295, C29×#12296, … |
| maximal quotient of | C29×#10649, C29×#10650, C29×#10651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 821);