Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C97 · Schur multiplier (H2) = C25 |
| direct product | C2×#471×C97 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C97 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C97 |
| prime radical | C2×AGL(1,ℤ/8)×C97 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C97 ◁ C22×C4×C97 ◁ G | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(4,8)(5,11)(6,14)(10,16)(12,13)(15,17);(3,4,6,8,7,10,14,16)(5,9,12,15,13,18,11,17);(3,5)(4,9)(6,12)(7,13)(8,15)(10,18)(11,14)(16,17)3104 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 1796);