Finite Group DatabasebetaFinite Group Databasebeta| order | 28800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C2×SO+(4,𝔽5) |
| derived | Ω+(4,𝔽5) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, socle, hypercenter | C22 | #72219548 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽5) | C22 |
| Frattini | C2 | C2×#72219548 |
| LCS | G ▷ Ω+(4,𝔽5) | |
| UCS | C1 ◁ C22 | |
1265 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 1265 conjugacy classes of subgroups — too many to draw. Only its 14 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 122 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 122 points:
(1,2)(3,17,45,66,99,15,9,43,62,48,115,82,77,64,42,121,107,108,102,21)(4,10,60,7,106,73,78,31,94,67,122,105,29,104,24,34,93,54,37,98)(5,81,50,57,103,59,96,88,95,28,119,19,76,86,55,49,18,14,33,83)(6,72,110,69,118,47,91,85,87,84,113,22,30,36,26,80,65,63,71,12)(8,27,11,20,92,70,23,44,58,111,116,112,100,101,32,89,90,25,109,38)(13,40,79,52,97,51,56,68,39,117,114,74,46,53,35,16,120,41,75,61);(3,90,39,61,101,42)(4,113,58,30,95,16)(5,98,112,10,22,79)(6,109,110,33,51,122)(7,15,74,57,8,54)(9,108,62,37,83,69)(11,41,12,29,13,25)(14,66,70,97,78,76)(17,19,18,47,53,32)(20,99,115,23,75,117)(21,87,26,38,59,56)(27,105,72,46,119,67)(28,36,107,43,102,94)(31,104,121,40,86,116)(34,45,65,73,77,91)(35,93,50,88,64,89)(44,100,68,84,60,114)(48,71,118,111,49,120)(52,92,82,81,96,80)80 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)(3,17,45,66,99,15,9,43,62,48,115,82,77,64,42,121,107,108,102,21)(4,10,60,7,106,73,78,31,94,67,122,105,29,104,24,34,93,54,37,98)(5,81,50,57,103,59,96,88,95,28,119,19,76,86,55,49,18,14,33,83)(6,72,110,69,118,47,91,85,87,84,113,22,30,36,26,80,65,63,71,12)(8,27,11,20,92,70,23,44,58,111,116,112,100,101,32,89,90,25,109,38)(13,40,79,52,97,51,56,68,39,117,114,74,46,53,35,16,120,41,75,61), (3,90,39,61,101,42)(4,113,58,30,95,16)(5,98,112,10,22,79)(6,109,110,33,51,122)(7,15,74,57,8,54)(9,108,62,37,83,69)(11,41,12,29,13,25)(14,66,70,97,78,76)(17,19,18,47,53,32)(20,99,115,23,75,117)(21,87,26,38,59,56)(27,105,72,46,119,67)(28,36,107,43,102,94)(31,104,121,40,86,116)(34,45,65,73,77,91)(35,93,50,88,64,89)(44,100,68,84,60,114)(48,71,118,111,49,120)(52,92,82,81,96,80));