Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23×C29 |
| prime radical | C29×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C29 ◁ C2×Q8×C29 ◁ G | |
216 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 56 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)(34,37)(35,40)(36,42)(38,48)(39,50)(43,58)(44,56)(46,54)(47,55)(49,59)(51,57)(52,60)(53,61);(30,32,35,44)(31,36,40,54)(33,46,48,42)(34,49,50,43)(37,52,39,53)(38,56,45,41)(47,60,51,61)(55,58,57,59);(31,37)(34,50)(36,52)(38,57)(39,40)(41,58)(43,49)(45,55)(47,51)(53,54)(56,59)(60,61)493 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#13679, C29×#13680, C2×C29×#2203, … |
| maximal normal subgroup of | C29×#13679, C29×#13680, C2×C29×#2203, … |
| maximal quotient of | C29×#12097, C29×#12098, C29×#12126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 935);