Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1902 |
| 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 | |
304 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 304 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)(32,41)(33,45,48,55)(34,49)(35,39)(36,40)(37,43)(38,47,54,46)(42,57,59,51)(44,52)(50,53)(56,58,61,60);(30,32,34,43,36,44,35,50)(31,37,39,41,40,53,49,52)(33,42,47,60,48,59,46,58)(38,51,55,56,54,57,45,61);(30,33)(31,38)(32,42)(34,47)(35,46)(36,48)(37,51)(39,55)(40,54)(41,56)(43,60)(44,59)(45,49)(50,58)(52,61)(53,57)841 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1902, C29×#20995, C29×#22284, … |
| maximal normal subgroup of | C2×C29×#1902, C29×#20995, C29×#22284, … |
| maximal quotient of | C29×#9087, C29×#9088, C29×#9091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 634);