Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C97 · Schur multiplier (H2) = C2 |
| direct product | C97×#1418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C97 | #357 |
| commutator | C16 | C2×C4×C97 |
| Frattini, cosocle | C2×C16 | C22×C97 |
| prime radical | #508×C97 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C97 ◁ C2×C4×C97 ◁ C2×C8×C97 ◁ C2×C16×C97 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 2 generators on 129 points:
(2,7,8,23)(3,11)(4,16)(5,18)(9,27,26,31)(10,28,20,25)(12,17,24,22)(13,19)(14,32)(15,21)(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);(1,2)(3,12)(4,17)(5,20)(6,8)(7,21)(9,16)(10,18)(11,23)(13,26)(14,27)(15,24)(19,22)(25,30)(28,32)(29,31)2522 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 150);