Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#72215280 |
| Fitting | C22×H(5,𝔽2) ⋊ D32 |
| derived | Ω+(4,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×S42 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽3) | C24 |
| Frattini | C2 | C22×S42 |
| Fitting | C22×H(5,𝔽2) | D32 |
| hypercenter | C23 | S42 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C23 | |
10106 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 10106 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2)(3,41,19)(5,6,46)(7,39,27)(8,44,21)(9,43,18)(10,17,47)(12,34,33)(13,26,29)(14,22,15)(16,37,50)(20,24,48)(23,40,25)(28,30,31)(35,38,45);(4,11)(12,28)(13,22)(14,29)(15,26)(16,35)(30,34)(31,33)(32,49)(36,42)(37,38)(45,50);(5,17)(6,8)(7,24)(9,23)(12,16)(13,22)(14,15)(18,43)(19,46)(20,48)(21,41)(25,40)(26,29)(27,39)(28,35)(30,42)(31,45)(32,38)(33,50)(34,36)(37,49)(44,47);(3,4)(5,16)(6,22)(7,15)(8,13)(9,26)(10,11)(12,17)(14,24)(18,42)(19,45)(20,38)(21,35)(23,29)(25,37)(27,36)(28,41)(30,43)(31,46)(32,48)(33,44)(34,39)(40,49)(47,50)112 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,41,19)(5,6,46)(7,39,27)(8,44,21)(9,43,18)(10,17,47)(12,34,33)(13,26,29)(14,22,15)(16,37,50)(20,24,48)(23,40,25)(28,30,31)(35,38,45), (4,11)(12,28)(13,22)(14,29)(15,26)(16,35)(30,34)(31,33)(32,49)(36,42)(37,38)(45,50), (5,17)(6,8)(7,24)(9,23)(12,16)(13,22)(14,15)(18,43)(19,46)(20,48)(21,41)(25,40)(26,29)(27,39)(28,35)(30,42)(31,45)(32,38)(33,50)(34,36)(37,49)(44,47), (3,4)(5,16)(6,22)(7,15)(8,13)(9,26)(10,11)(12,17)(14,24)(18,42)(19,45)(20,38)(21,35)(23,29)(25,37)(27,36)(28,41)(30,43)(31,46)(32,48)(33,44)(34,39)(40,49)(47,50));