Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C2 |
| direct product | C29×#1587 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C42 | C23×C29 |
| socle, prime radical | C22×C29 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C4×C8×C29 ◁ G | |
152 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 152 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) = 53 |
| transitive degree | μt(G) = 3712 |
A minimal faithful permutation representation — 3 generators on 53 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)(30,31,33,32,34,35,37,36)(38,39)(40,46)(41,49)(42,44)(43,47)(45,51)(48,53)(50,52);(38,40,42,48)(39,43,44,52)(41,50,51,47)(45,53,49,46);(30,32,33,35,34,36,37,31)(39,44)(43,52)(45,49)(46,53)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C29×#1587, C29×#17868, C29×#17870, … |
| maximal normal subgroup of | C2×C29×#1587, C29×#17868, C29×#17870, … |
| maximal quotient of | C29×#8205, C29×#8213, C29×#8288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 319);