Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#1482 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C42 | C23×C29 |
| socle | C22×C29 | C2×C22⋊C4 |
| prime radical | C29×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C2×C42×C29 ◁ G | |
212 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 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)(32,39)(33,37)(34,42)(35,38)(36,40)(41,44)(43,45)(46,47,49,52,50,53,56,59)(48,51,55,58,54,57,60,61);(30,32)(31,36)(33,40)(34,43)(35,41)(37,44)(38,45)(39,42)(46,48)(47,51)(49,55)(50,54)(52,58)(53,57)(56,60)(59,61);(30,33,35,34)(31,37,38,42)(32,40,41,43)(36,44,45,39)(48,54)(51,57)(55,60)(58,61)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#8489, C29×#8490, C29×#8514, … |
| maximal normal subgroup of | C29×#8489, C29×#8490, C29×#8514, … |
| maximal quotient of | C29×#7656, C29×#7657, C29×#7658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 214);