Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#11711587 |
| Fitting | C2×#1791 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020⋊C2 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C22×C4 | C22×S4 |
| Fitting | C2×#1791 | D3 |
| socle | C22 | #1020⋊C4 |
| prime radical | C2×#87519 | C2 |
| cosocle | #5950 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 | |
1517 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1517 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2)(3,20,8)(4,29,13)(5,19,16)(6,32,18)(9,22,23)(10,28,25)(11,34,27)(14,33,31);(3,5,6,14,9,4,10,11)(7,19,17,33,21,29,24,34)(8,13,18,27,23,16,25,31)(12,28,26,20,15,32,30,22);(3,4,6,11,9,5,10,14)(7,15,17,30,21,12,24,26)(8,16,18,31,23,13,25,27)(19,32,33,22,29,28,34,20)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72341881, #72341905, C2×#72236904, … |
| maximal normal subgroup of | #72341881, #72341905, C2×#72236904, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,20,8)(4,29,13)(5,19,16)(6,32,18)(9,22,23)(10,28,25)(11,34,27)(14,33,31), (3,5,6,14,9,4,10,11)(7,19,17,33,21,29,24,34)(8,13,18,27,23,16,25,31)(12,28,26,20,15,32,30,22), (3,4,6,11,9,5,10,14)(7,15,17,30,21,12,24,26)(8,16,18,31,23,13,25,27)(19,32,33,22,29,28,34,20));