Finite Group DatabasebetaFinite Group Databasebeta| order | 43008 |
| isoclinism | isoclinic class of #72227552 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini | C25 | AGL(3,𝔽2) |
| Fitting, cosocle | C2×#2867 | GL(3,𝔽2) |
| socle | C23 | #72209612 |
| LCS | G | |
| UCS | C1 | |
422 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 422 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) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,49,115,65,14,38,122,95,22,76,99,68,4,75)(2,25,112,33,96,123,63,87,9,82,32,46,74,39)(3,12,85,11,28,72,36,118,55,104,86,93,117,30)(5,78,18,64,15,91,17,35,108,59,61,67,37,70)(6,84,73,51,103,90,128,29,19,47,101,54,42,71)(7,21,121,13,23,26,110,97,50,57,102,53,24,44)(8,83,113,88,66,94,106,41,10,45,120,16,27,111)(20,125,69,80,89,79,81,107,116,40,48,126,62,109)(31,98,60,127,56,92,58,34,77,114,119,52,43,124)(100,105);(1,36,35,34,33,7)(2,3,125,16,98,40)(4,17,114,27,72,59)(5,29,95,97,41,31)(6,8)(9,122,60,37,128,18)(10,100,15,78,109,68)(11,19,126,54,50,107)(12,26,70,45,93,117)(13,88,104)(14,21,81,67,84,105)(20,103,83,89,86,108)(22,79,56,46,80,53)(23,118,48,52,77,62)(24,44,115,28,112,121)(25,58,73,91,111,123)(30,32)(38,110,47,94,82,74)(39,64,101)(42,75,113,92,71,55)(43,106,57,90,124,76)(49,69,66,96,116,87)(61,63,119)(85,99,102)32 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((1,49,115,65,14,38,122,95,22,76,99,68,4,75)(2,25,112,33,96,123,63,87,9,82,32,46,74,39)(3,12,85,11,28,72,36,118,55,104,86,93,117,30)(5,78,18,64,15,91,17,35,108,59,61,67,37,70)(6,84,73,51,103,90,128,29,19,47,101,54,42,71)(7,21,121,13,23,26,110,97,50,57,102,53,24,44)(8,83,113,88,66,94,106,41,10,45,120,16,27,111)(20,125,69,80,89,79,81,107,116,40,48,126,62,109)(31,98,60,127,56,92,58,34,77,114,119,52,43,124)(100,105), (1,36,35,34,33,7)(2,3,125,16,98,40)(4,17,114,27,72,59)(5,29,95,97,41,31)(6,8)(9,122,60,37,128,18)(10,100,15,78,109,68)(11,19,126,54,50,107)(12,26,70,45,93,117)(13,88,104)(14,21,81,67,84,105)(20,103,83,89,86,108)(22,79,56,46,80,53)(23,118,48,52,77,62)(24,44,115,28,112,121)(25,58,73,91,111,123)(30,32)(38,110,47,94,82,74)(39,64,101)(42,75,113,92,71,55)(43,106,57,90,124,76)(49,69,66,96,116,87)(61,63,119)(85,99,102));