Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C97 · Schur multiplier (H2) = C22 |
| direct product | C97×#1381 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C97 | D8 |
| commutator | C8 | C42×C97 |
| Frattini, cosocle | C22×C8 | C22×C97 |
| socle | C22×C97 | #107 |
| prime radical | C23×C97 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C97 ◁ C22×C4×C97 ◁ C22×C8×C97 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 2 generators on 133 points:
(5,6,9,14,11,17,23,16)(7,18,20,32,22,33,34,27)(8,15,10,13,24,30,25,31)(12,21,26,35,29,36,28,19)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(1,2,3,4)(5,7,10,21,9,20,24,35,11,22,25,36,23,34,8,19)(6,12,15,27,16,28,31,33,17,29,30,32,14,26,13,18)4268 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 113);