Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C97 · Schur multiplier (H2) = C1 |
| direct product | C97×#1398 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C97 | C42 |
| commutator | C4 | C4×C8×C97 |
| Frattini, cosocle | C2×C16 | C22×C97 |
| socle | C2×C97 | #346 |
| prime radical | C22×C97 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8×C97 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 2 generators on 129 points:
(1,2,3,8,4,10,13,23,5,11,14,21,15,25,27,31,6,9,12,22,16,24,26,32,17,7,18,19,28,20,29,30)(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);(2,7,9,11)(3,12)(8,21,22,19)(10,20,24,25)(13,26)(14,18)(23,31,32,30)(27,29)4268 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 130);