Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
| direct product | C22×#87521 |
| Fitting | C22×#585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#1020 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C2 | C23×#1020 |
| Fitting | C22×#585 | D3 |
| hypercenter | C22×C4 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
13884 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 13884 conjugacy classes of subgroups — too many to draw. Only its 158 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)(5,13,8)(6,19,12)(9,20,15)(10,18,17);(1,2)(3,4)(5,6,9,10)(7,16,14,11)(8,17,15,12)(13,19,20,18);(1,2);(6,10)(7,13)(11,18)(12,17)(14,20)(16,19)92 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#11715839, C22×#11714184, C2×#72236781, … |
| maximal normal subgroup of | C22×#11715839, C22×#11714184, C2×#72236781, … |
| maximal quotient of | C2×#72240901, C2×#72245747 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,13,8)(6,19,12)(9,20,15)(10,18,17), (1,2)(3,4)(5,6,9,10)(7,16,14,11)(8,17,15,12)(13,19,20,18), (1,2), (6,10)(7,13)(11,18)(12,17)(14,20)(16,19));