Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C2×C4 |
| direct product | C29×#1473 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C24×C29 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C2×C42×C29 ◁ G | |
214 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 88 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)(30,31,33,38,35,39,45,52)(32,37,42,50,40,51,53,61)(34,36,44,47,41,48,55,59)(43,46,57,58,54,49,56,60);(31,36)(32,40)(33,44)(37,49)(39,48)(42,56)(43,54)(45,55)(46,51)(50,61)(53,57)(58,60);(30,32,41,54)(31,37,48,49)(33,42,55,56)(34,43,35,40)(36,46,39,51)(38,50,59,60)(44,57,45,53)(47,58,52,61)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#8489, C29×#8490, C2×C29×#1473, … |
| maximal normal subgroup of | C29×#8489, C29×#8490, C2×C29×#1473, … |
| maximal quotient of | C29×#7666, C29×#7669, C29×#7672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 205);