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) = C23 |
| direct product | C29×#2135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | C2×#99 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C2×Q8 | 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 | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 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)(30,31,35,40,36,41,33,38)(32,42,45,60,46,57,43,59)(34,48,47,58,50,39,51,56)(37,52,53,49,54,61,55,44);(30,32,36,46)(31,37,41,54)(33,45,35,43)(34,49,50,44)(38,55,40,53)(39,57,48,42)(47,52,51,61)(56,59,58,60);(30,33,36,35)(31,38,41,40)(32,43,46,45)(34,47,50,51)(37,53,54,55)(39,56,48,58)(42,59,57,60)(44,52,49,61)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×#11450, C29×#11481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 867);