Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C101 · Schur multiplier (H2) = C27 |
| direct product | C4×B(4,𝔽2)×C101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C101 | #120 |
| commutator | C23 | C23×C4×C101 |
| Frattini, cosocle | C24 | C24×C101 |
| socle | C22×C101 | C2×#120 |
| prime radical | C2×B(4,𝔽2)×C101 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C101 ◁ C23×C4×C101 ◁ G | |
1910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 3232 |
A minimal faithful permutation representation — 4 generators on 113 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(9,12);(5,6)(7,9)(8,10)(11,12);(5,7)(6,9)(8,11)(10,12)6464 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 14467);