Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #386 |
| commutator | C23 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C4×C29 ◁ G | |
308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 308 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)(30,31,34,39,35,38,40,53)(32,36,43,50,42,51,47,54)(33,44,46,55,45,37,48,52)(41,49,57,60,56,59,58,61);(30,32,40,43,35,42,34,47)(31,36,39,54,38,51,53,50)(33,41,46,58,45,56,48,57)(37,49,55,60,44,59,52,61);(30,33)(31,37)(32,41)(34,48)(35,45)(36,49)(38,44)(39,55)(40,46)(42,56)(43,58)(47,57)(50,61)(51,59)(52,53)(54,60)841 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12757, C29×#12758, C29×#12770, … |
| maximal normal subgroup of | C29×#12757, C29×#12758, C29×#12770, … |
| maximal quotient of | C29×#9108, C29×#9110, C29×#9189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 647);