Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C22×C4 |
| direct product | C29×#2199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C29 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C29 ◁ C2×D4×C29 ◁ G | |
270 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 270 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) = 45 |
| transitive degree | μt(G) = 464 |
A minimal faithful permutation representation — 3 generators on 45 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);(30,32)(31,35)(33,40)(34,39)(36,38)(37,42)(41,45)(43,44);(32,39)(33,41)(35,42)(38,44)493 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#13677, C29×#13678, C2×C29×#2199, … |
| maximal normal subgroup of | C29×#13677, C29×#13678, C2×C29×#2199, … |
| maximal quotient of | C29×#12093, C29×#12100, C29×#12123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 931);