Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C97 · Schur multiplier (H2) = C22 |
| direct product | C97×#2176 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C97 | C2×D8 |
| commutator | C8 | C22×C4×C97 |
| Frattini, cosocle | C2×C8 | C23×C97 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C97 ◁ C42×C97 ◁ C4×C8×C97 ◁ G | |
148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 3 generators on 133 points:
(1,2,4,3)(5,6)(7,15)(8,19)(9,24)(10,12)(11,16)(13,22)(14,21)(17,35)(18,32)(20,30)(23,31)(25,29)(26,34)(27,36)(28,33)(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);(5,7,10,18)(6,11,12,26)(8,20,23,36)(9,25,21,17)(13,29,28,35)(14,30,24,27)(15,31,32,19)(16,33,34,22);(1,3,4,2)(6,12)(11,26)(13,28)(14,24)(15,32)(19,31)(27,30)(29,35)3104 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 908);