Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#1814 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #121 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C2×C29 | #380 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C22×C4×C29 ◁ G | |
256 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 116 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)(33,46,48,53)(34,49)(35,39)(36,40)(37,43)(38,47,54,50)(42,51,58,61)(44,52)(45,56)(55,57,60,59);(30,32,34,43,36,45,35,44)(31,37,39,41,40,52,49,56)(33,42,47,57,48,58,50,59)(38,51,53,55,54,61,46,60);(30,33)(31,38)(32,42)(34,47)(35,50)(36,48)(37,51)(39,53)(40,54)(41,55)(43,57)(44,59)(45,58)(46,49)(52,61)(56,60)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12747, C29×#12750, C29×#12751, … |
| maximal normal subgroup of | C29×#12747, C29×#12750, C29×#12751, … |
| maximal quotient of | C29×#8604, C29×#8605, C29×#8606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 546);