Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C25 |
| direct product | C29×#1796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #379 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×C29×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C4×C29 ◁ G | |
536 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 132 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)(33,44,46,37)(34,47)(35,39)(36,42)(38,49)(41,55,57,54)(43,53)(45,52,61,59)(48,51)(50,56,58,60);(30,32,35,43)(31,36,39,51)(33,41,46,57)(34,48,49,42)(37,50,44,58)(38,53,47,40)(45,60,61,56)(52,55,59,54);(30,33)(31,37)(32,41)(34,45)(35,46)(36,50)(38,52)(39,44)(40,54)(42,56)(43,57)(47,59)(48,60)(49,61)(51,58)(53,55)841 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12712, C29×#12714, C29×#12715, … |
| maximal normal subgroup of | C29×#12712, C29×#12714, C29×#12715, … |
| maximal quotient of | C29×#8323, C29×#8324, C29×#8325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 528);