Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#11711589 |
| 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 | |
1481 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1481 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:
(3,6,17,7)(4,16,25,19)(5,13,28,14)(8,18,31,20)(9,11,30,12)(10,27,32,29)(15,24,33,26)(21,22,34,23);(3,4,5,10,8,9,15,21)(6,30,14,27,31,12,24,32,7,16,28,29,18,25,26,22,17,19,13,34,20,11,33,23);(1,2)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72236892, C2×#72236895, C2×#72236849, … |
| maximal normal subgroup of | C2×#72236892, C2×#72236895, C2×#72236849, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,17,7)(4,16,25,19)(5,13,28,14)(8,18,31,20)(9,11,30,12)(10,27,32,29)(15,24,33,26)(21,22,34,23), (3,4,5,10,8,9,15,21)(6,30,14,27,31,12,24,32,7,16,28,29,18,25,26,22,17,19,13,34,20,11,33,23), (1,2));