Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C97 · Schur multiplier (H2) = C2×C4 |
| direct product | C97×#1384 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C97 | #106 |
| commutator | C8 | C42×C97 |
| Frattini, cosocle | C22×C8 | C22×C97 |
| socle | C2×C97 | #340 |
| prime radical | C23×C97 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C97 ◁ C22×C4×C97 ◁ C22×C8×C97 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 2 generators on 129 points:
(1,2)(3,13,15,27)(4,10)(5,20)(6,22)(7,12)(8,14,23,28)(9,18)(11,21)(16,31,29,25)(17,32,30,24)(19,26)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(1,3,5,16,21,30,19,28,7,15,18,29,6,17,4,14)(2,8,10,24,22,31,9,13,12,23,26,32,11,25,20,27)3104 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 116);