Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| direct product | C2×2−1+4⋊A5 |
| Fitting | C2×2−1+4 ⋊ A5 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #11757163 |
| commutator, hypocenter | 2−1+4⋊A5 | C2 |
| Frattini | C2 | C2×#11757163 |
| Fitting | C2×2−1+4 | A5 |
| cosocle | 2−1+4 | C2×A5 |
| LCS | G ▷ 2−1+4⋊A5 | |
| UCS | C1 ◁ C22 | |
341 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 9 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 — 2 generators on 34 points:
(1,2)(3,6,23,31,22,9)(4,10,19,29,14,5)(7,13)(8,32)(11,18)(12,33)(15,28,24,17,20,34)(16,26,27,30,25,21);(5,21,13)(6,24,26)(7,8,23)(9,28,18)(10,16,17)(11,12,19)(15,34,29)(25,30,31)32 conjugacy classes — view the full character table.
| maximal subgroup of | #72267138, #72267349, #72269016, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,6,23,31,22,9)(4,10,19,29,14,5)(7,13)(8,32)(11,18)(12,33)(15,28,24,17,20,34)(16,26,27,30,25,21), (5,21,13)(6,24,26)(7,8,23)(9,28,18)(10,16,17)(11,12,19)(15,34,29)(25,30,31));