Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#1868 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #120 |
| commutator | C23 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C2×C29 | #385 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C22×C4×C29 ◁ G | |
260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 108 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)(30,31,34,38,35,39,45,51)(32,36,42,48,41,47,55,59)(33,37,44,50,43,49,56,60)(40,52,54,61,53,46,57,58);(30,32,34,42,35,41,45,55)(31,36,38,48,39,47,51,59)(33,40,44,54,43,53,56,57)(37,46,50,58,49,52,60,61);(30,33)(31,37)(32,40)(34,44)(35,43)(36,46)(38,50)(39,49)(41,53)(42,54)(45,56)(47,52)(48,58)(51,60)(55,57)(59,61)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12962, C29×#12963, C29×#12968, … |
| maximal normal subgroup of | C29×#12962, C29×#12963, C29×#12968, … |
| maximal quotient of | C29×#8851, C29×#8852, C29×#8861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 600);