Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C97 · Schur multiplier (H2) = C2×C4 |
| direct product | C97×#2158 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C97 | D8 |
| commutator | C8 | C22×C4×C97 |
| Frattini, cosocle | C2×C8 | C23×C97 |
| socle | C22×C97 | C2×D8 |
| prime radical | C23×C97 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C97 ◁ C22×C4×C97 ◁ C22×C8×C97 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 3 generators on 133 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,23)(10,25)(11,15)(12,19)(14,17)(18,29)(20,35)(21,31)(22,28)(24,36)(26,33)(27,34)(30,32)(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);(5,7,17,32,10,20,24,34,11,21,22,33,8,18,9,19)(6,12,23,29,13,26,28,31,15,27,36,35,25,30,14,16);(5,8,11,10)(6,13,15,25)(7,18,21,20)(9,22,24,17)(12,26,27,30)(14,23,28,36)(16,29,31,35)(19,33,34,32)4268 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 890);