Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11715681 |
| Fitting | C2×#3573 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | #1020 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C4 | C22×#1020 |
| Fitting | C2×#3573 | D3 |
| socle | C22 | C4×#1020 |
| prime radical | C2×#87520 | C2 |
| cosocle | #5950 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C8 | |
2833 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 2833 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(3,26,27,22,19,23,15,21,16,12,14,20,8,11,13,9,7,10,5,34,6,4,30,31)(17,25,28,33,18,24,29,32);(1,2)(6,16)(10,23)(13,27)(17,18)(20,31)(24,25)(28,29)(32,33);(6,16)(7,18)(10,23)(11,25)(13,27)(14,29)(17,19)(20,31)(21,33)(24,26)(28,30)(32,34)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72227323, #72227328, C2×#72228092, … |
| maximal normal subgroup of | #72227323, #72227328, #72341722, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,26,27,22,19,23,15,21,16,12,14,20,8,11,13,9,7,10,5,34,6,4,30,31)(17,25,28,33,18,24,29,32), (1,2)(6,16)(10,23)(13,27)(17,18)(20,31)(24,25)(28,29)(32,33), (6,16)(7,18)(10,23)(11,25)(13,27)(14,29)(17,19)(20,31)(21,33)(24,26)(28,30)(32,34));