Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#2987 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C24 |
| commutator | C22 | C23×C4×C29 |
| Frattini, cosocle | C2×C4 | C24×C29 |
| socle | C22×C29 | C23×C4 |
| prime radical | C23×C29 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ G | |
314 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 1856 |
A minimal faithful permutation representation — 4 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)(32,38)(33,36)(34,37)(35,40)(39,44)(41,43)(42,45)(46,47,49,53,50,52,56,59)(48,51,55,58,54,57,60,61);(30,32,34,40)(31,35,37,38)(33,41,42,39)(36,44,45,43)(46,48)(47,51)(49,55)(50,54)(52,57)(53,58)(56,60)(59,61);(30,33,34,42)(31,36,37,45)(32,39,40,41)(35,43,38,44)(47,52)(48,54)(53,59)(55,60);(31,37)(35,38)(36,45)(43,44)(48,54)(51,57)(55,60)(58,61)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#8919, C29×#9006, C29×#9008, … |
| maximal normal subgroup of | C29×#8919, C29×#9006, C29×#9008, … |
| maximal quotient of | C29×#20539, C29×#20542, C29×#20576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1719);