Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C29 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C2×C4×C29 ◁ C29×#131 ◁ G | |
220 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 64 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)(32,43)(33,47)(34,41)(35,40)(36,42)(37,44)(38,45)(39,50)(46,58)(48,59)(49,53)(51,57)(52,56)(54,61)(55,60);(30,32)(31,37)(33,48)(34,45)(35,52)(36,46)(38,53)(39,56)(40,54)(41,59)(42,55)(43,47)(44,50)(49,60)(51,61)(57,58);(30,32)(31,38)(33,44)(34,52)(35,45)(36,46)(37,53)(39,55)(40,58)(41,43)(42,56)(47,59)(48,50)(49,61)(51,60)(54,57)667 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#2192, C29×#33538, C29×#33539, … |
| maximal normal subgroup of | C2×C29×#2192, C29×#33538, C29×#33539, … |
| maximal quotient of | C29×#12108, C29×#12110, C29×#12164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 924);