Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#2212 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C29 |
| prime radical | C29×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C2×C4×C29 ◁ C4×C8×C29 ◁ G | |
154 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 60 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)(33,45)(34,49)(35,51)(36,41)(37,43)(38,42)(39,50)(44,58)(46,57)(47,61)(48,53)(52,54)(55,59)(56,60);(30,32,34,43,36,44,50,35)(31,37,39,40,41,51,49,58)(33,42,46,59,48,60,47,52)(38,53,54,57,56,45,55,61);(30,33)(31,38)(32,42)(34,46)(35,52)(36,48)(37,53)(39,54)(40,57)(41,56)(43,59)(44,60)(45,51)(47,50)(49,55)(58,61)667 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#2212, C29×#33594, C29×#33595, … |
| maximal normal subgroup of | C2×C29×#2212, C29×#33594, C29×#33595, … |
| maximal quotient of | C29×#12416, C29×#12417, C29×#12420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 944);