Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C101 · Schur multiplier (H2) = C22×C4 |
| direct product | C101×#7916 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C101 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C101 |
| Frattini, cosocle | C23×C4 | C23×C101 |
| prime radical | C24×C101 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C101 ◁ C22×C42×C101 ◁ G | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 12928 |
A minimal faithful permutation representation — 3 generators on 137 points:
(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22,24,28,25,27,33,35)(23,29,31,36,30,26,32,34)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,26)(24,32)(25,30)(27,29)(28,36)(31,33)(34,35);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(23,30)(26,29)(31,32)(34,36)6464 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 904);