Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C29 · Schur multiplier (H2) = C4 |
| direct product | C29×#2164 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C29 | C23 |
| commutator | C22 | C22×C8×C29 |
| Frattini, cosocle | C2×C8 | C23×C29 |
| socle | C22×C29 | C22×C8 |
| prime radical | C23×C29 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C29 ◁ G | |
138 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 69 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)(32,35)(33,34)(36,37)(38,39,41,45,42,46,51,57,43,47,52,58,53,59,63,67)(40,44,49,55,50,56,62,66,48,54,60,64,61,65,68,69);(31,34)(35,37)(38,40)(39,44)(41,49)(42,50)(43,48)(45,55)(46,56)(47,54)(51,62)(52,60)(53,61)(57,66)(58,64)(59,65)(63,68)(67,69);(30,32,33,36)(31,35,34,37)(40,48)(44,54)(49,60)(50,61)(55,64)(56,65)(62,68)(66,69)1624 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C29×#2164, C29×#33419, C29×#33420, … |
| maximal normal subgroup of | C2×C29×#2164, C29×#33419, C29×#33420, … |
| maximal quotient of | C29×#12592, C29×#12600, C29×#12654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 896);