Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C25 |
| direct product | C29×#1794 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #120 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C2×C29 | #379 |
| prime radical | C29×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C22×C4×C29 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 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)(30,31,35,39,36,40,34,47)(32,41,43,37,44,52,48,51)(33,38,45,54,46,55,49,53)(42,57,60,61,59,56,58,50);(30,32)(31,37)(33,42)(34,48)(35,43)(36,44)(38,50)(39,52)(40,51)(41,47)(45,60)(46,59)(49,58)(53,56)(54,57)(55,61);(30,33)(31,38)(32,42)(34,49)(35,45)(36,46)(37,50)(39,54)(40,55)(41,56)(43,60)(44,59)(47,53)(48,58)(51,61)(52,57)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#7346, C29×#7347, C29×#13045, … |
| maximal normal subgroup of | C29×#7346, C29×#7347, C29×#13045, … |
| maximal quotient of | C29×#8318, C29×#8319, C29×#8320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 526);