Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C97 · Schur multiplier (H2) = C26 |
| direct product | C97×#3005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C97 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C97 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C97 ◁ C23×C4×C97 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 4 generators on 117 points:
(1,2)(3,4)(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(2,4)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14);(6,13)(11,20)(12,17)(14,19)3104 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 1737);