Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1909 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #386 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C4×C29 ◁ G | |
312 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 312 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)(32,40)(33,44,46,37)(34,47)(35,39)(36,42)(38,49)(41,54,57,61)(43,55)(45,52,48,53)(50,56,60,58)(51,59);(30,32,34,42,35,43,49,59)(31,36,38,40,39,51,47,55)(33,41,48,58,46,57,45,56)(37,50,53,54,44,60,52,61);(30,33)(31,37)(32,41)(34,48)(35,46)(36,50)(38,53)(39,44)(40,54)(42,58)(43,57)(45,49)(47,52)(51,60)(55,61)(56,59)841 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12756, C29×#12757, C29×#12779, … |
| maximal normal subgroup of | C29×#12756, C29×#12757, C29×#12779, … |
| maximal quotient of | C29×#9099, C29×#9100, C29×#9103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 641);