Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C3×SL(2,𝔽5) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S5 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽5) | C22 |
| Fitting, socle | C2×C3 | C2×S5 |
| LCS | G ▷ C3×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 | |
170 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
Permutation generators:
(1,4)(2,3)(5,29)(6,30)(7,31)(8,32)(11,12)(13,36)(14,33)(15,34)(16,35)(17,20)(18,19)(21,23)(22,24)(25,37)(26,38)(27,40)(28,39)(41,42,43);(1,32,35,40,5,3,29,33,39,8,2,31,34,38,6,4,30,36,37,7)(9,28,20,22,14,11,27,17,23,16,10,25,19,21,13,12,26,18,24,15)(41,43)30 conjugacy classes — view the full character table.
| maximal subgroup of | SO+(4,𝔽5), C2×#13088295 |
| maximal normal subgroup of | C2×#13088295 |
| maximal quotient of | C2×#13088295 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4612);