Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#11711588 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020⋊C2 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C22×C4 | C22×S4 |
| Fitting | C2×#3025 | D3 |
| socle | C22 | #73080 |
| prime radical | C2×#3025⋊C3 | C2 |
| cosocle | #5950 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 | |
1277 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 1277 conjugacy classes of subgroups — too many to draw. Only its 67 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)(3,16,21)(4,20,28)(5,27,32)(6,18,8)(9,31,34)(10,26,12)(13,30,15)(22,33,24);(3,4,5,9,8,12,15,24)(6,17,13,29,16,11,27,23)(7,20,14,31,19,10,25,22)(18,28,30,34,21,26,32,33);(6,16)(7,19)(10,20)(11,17)(13,27)(14,25)(18,21)(22,31)(23,29)(26,28)(30,32)(33,34)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72236846, #72342061, #72342800, … |
| maximal normal subgroup of | C2×#72236846, #72342061, #72342800, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,16,21)(4,20,28)(5,27,32)(6,18,8)(9,31,34)(10,26,12)(13,30,15)(22,33,24), (3,4,5,9,8,12,15,24)(6,17,13,29,16,11,27,23)(7,20,14,31,19,10,25,22)(18,28,30,34,21,26,32,33), (6,16)(7,19)(10,20)(11,17)(13,27)(14,25)(18,21)(22,31)(23,29)(26,28)(30,32)(33,34));