Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C22×#73025 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C23 | C23×S4 |
| Fitting | C22×B(4,𝔽2) | D3 |
| prime radical | C22×B(4,𝔽2)⋊C3 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C23 | |
3740 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 3740 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(3,4)(5,8,16)(9,15,19)(10,20,12)(11,14,18);(5,6,9,13)(7,14,17,12)(8,18,15,10)(11,16,20,19);(11,20)(16,19);(1,2)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72341870, C22×#11714162, C2×#72236902, … |
| maximal normal subgroup of | #72341870, C22×#11714162, C2×#72236902, … |
| maximal quotient of | C2×#72245696 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,8,16)(9,15,19)(10,20,12)(11,14,18), (5,6,9,13)(7,14,17,12)(8,18,15,10)(11,16,20,19), (11,20)(16,19), (1,2));