Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C3 |
| direct product | C2×#72224794 |
| Fitting | C2×H(7,𝔽2) ⋊ C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×A4⋊S4 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽3) | C24 |
| Frattini | C2 | C23×A4⋊S4 |
| Fitting | C2×H(7,𝔽2) | C3⋊D3 |
| hypercenter | C2×D4 | A4⋊S4 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
12765 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 12765 conjugacy classes of subgroups — too many to draw. Only its 230 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:
(3,15,22,11,8,32,10,13,23,4,6,31)(5,29,25,26,19,28,21,44,18,12,41,34)(7,14,9,16)(17,27,37,43)(20,30,24,33)(35,50,39,47,42,49,38,46,40,45,36,48);(1,2)(3,5,17)(4,12,27)(6,20,36)(7,18,39)(8,24,42)(9,25,40)(10,21,37)(11,26,43)(13,30,46)(14,28,48)(15,33,50)(16,34,49)(19,35,22)(23,41,38)(29,45,31)(32,44,47);(5,17)(6,8)(7,23)(9,22)(12,27)(13,15)(14,32)(16,31)(18,38)(19,40)(20,42)(21,37)(24,36)(25,35)(26,43)(28,47)(29,49)(30,50)(33,46)(34,45)(39,41)(44,48);(4,11)(12,26)(13,15)(14,16)(27,43)(28,34)(29,44)(30,33)(31,32)(45,47)(46,50)(48,49)124 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((3,15,22,11,8,32,10,13,23,4,6,31)(5,29,25,26,19,28,21,44,18,12,41,34)(7,14,9,16)(17,27,37,43)(20,30,24,33)(35,50,39,47,42,49,38,46,40,45,36,48), (1,2)(3,5,17)(4,12,27)(6,20,36)(7,18,39)(8,24,42)(9,25,40)(10,21,37)(11,26,43)(13,30,46)(14,28,48)(15,33,50)(16,34,49)(19,35,22)(23,41,38)(29,45,31)(32,44,47), (5,17)(6,8)(7,23)(9,22)(12,27)(13,15)(14,32)(16,31)(18,38)(19,40)(20,42)(21,37)(24,36)(25,35)(26,43)(28,47)(29,49)(30,50)(33,46)(34,45)(39,41)(44,48), (4,11)(12,26)(13,15)(14,16)(27,43)(28,34)(29,44)(30,33)(31,32)(45,47)(46,50)(48,49));