Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C22×C4 |
| direct product | C29×#2129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | C2×#99 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C2×D4 | C23×C29 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C2×C4×C29 ◁ C2×#41×C29 ◁ G | |
238 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 84 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)(32,40,43,55)(33,37)(34,47,49,38)(35,39)(36,42)(41,56,44,58)(45,60,61,53)(46,54)(48,51)(50,57)(52,59);(30,32,35,43)(31,36,39,51)(33,44,46,41)(34,48,49,42)(37,52,54,50)(38,55,47,40)(45,57,61,59)(53,56,60,58);(30,33,35,46)(31,37,39,54)(32,41,43,44)(34,45,49,61)(36,50,51,52)(38,53,47,60)(40,56,55,58)(42,57,48,59)667 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#12712, C29×#12713, C29×#12721, … |
| maximal normal subgroup of | C29×#12712, C29×#12713, C29×#12721, … |
| maximal quotient of | C29×#11434, C29×#11449, C29×#11482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 861);