Finite Group DatabasebetaFinite Group Databasebeta| order | 12000 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| affine general linear | AGL(2,𝔽5) |
| Fitting, socle | C52 ⋊ GL(2,𝔽5) |
| derived | #72209598 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | #72209598 | C4 |
| Fitting, socle | C52 | GL(2,𝔽5) |
| prime radical, cosocle | #72210652 | C2 |
| LCS | G ▷ #72209598 | |
| UCS | C1 | |
132 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 7 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(2,6,22,8)(3,9,29,7)(4,13,34,15)(5,18,16,14)(10,25,49,31)(12,23,47,48)(17,39,38,43)(19,37,45,46)(21,35,40,42)(26,30,32,27);(2,7,26,12,31)(3,10,30,6,23)(4,14,38,21,46)(5,19,43,13,35)(8,27,49,29,48)(9,22,25,47,32)(15,39,45,16,42)(17,18,34,37,40);(2,8,6,22,24)(3,11,29,9,7)(4,15,13,34,36)(5,20,16,18,14)(12,33,47,23,48)(17,39,41,43,38)(21,44,40,35,42)(26,32,27,28,30);(3,9,7,11)(5,18,14,20)(6,24,22,8)(10,31,49,25)(12,23,48,33)(13,36,34,15)(17,41,39,38)(19,46,45,37)(21,35,42,44)(26,32,28,27);(3,12,27,22)(5,21,39,34)(6,25,26,33)(7,10,8,28)(9,24,48,31)(11,32,49,23)(13,37,38,44)(14,19,15,41)(17,45,35,20)(18,36,42,46);(1,4,16,43,40)(5,17,44,46,15)(13,18,41,21,45)(14,39,35,37,36)(19,34,20,38,42);(1,5,21,34,39)(4,17,45,20,35)(13,38,37,16,44)(14,40,15,41,19)(18,42,36,43,46)29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(2,𝔽5) |
| maximal normal subgroup of | C2×AGL(2,𝔽5) |
| maximal quotient of | C2×AGL(2,𝔽5) |
| automorphism group of | C5⋊D5, #1043, H(3,𝔽5), … |
| holomorph of | C52 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((2,6,22,8)(3,9,29,7)(4,13,34,15)(5,18,16,14)(10,25,49,31)(12,23,47,48)(17,39,38,43)(19,37,45,46)(21,35,40,42)(26,30,32,27), (2,7,26,12,31)(3,10,30,6,23)(4,14,38,21,46)(5,19,43,13,35)(8,27,49,29,48)(9,22,25,47,32)(15,39,45,16,42)(17,18,34,37,40), (2,8,6,22,24)(3,11,29,9,7)(4,15,13,34,36)(5,20,16,18,14)(12,33,47,23,48)(17,39,41,43,38)(21,44,40,35,42)(26,32,27,28,30), (3,9,7,11)(5,18,14,20)(6,24,22,8)(10,31,49,25)(12,23,48,33)(13,36,34,15)(17,41,39,38)(19,46,45,37)(21,35,42,44)(26,32,28,27), (3,12,27,22)(5,21,39,34)(6,25,26,33)(7,10,8,28)(9,24,48,31)(11,32,49,23)(13,37,38,44)(14,19,15,41)(17,45,35,20)(18,36,42,46), (1,4,16,43,40)(5,17,44,46,15)(13,18,41,21,45)(14,39,35,37,36)(19,34,20,38,42), (1,5,21,34,39)(4,17,45,20,35)(13,38,37,16,44)(14,40,15,41,19)(18,42,36,43,46));