Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C97 · Schur multiplier (H2) = C2 |
| direct product | C97×#1413 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C97 | #354 |
| commutator | C42 | C2×C4×C97 |
| Frattini, cosocle | C4⋊Q8 | C22×C97 |
| prime radical | C22×C97 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C97 ◁ C22×C97 ◁ C2×C4×C97 ◁ C4⋊Q8×C97 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 2 generators on 129 points:
(1,2,4,9,6,11,18,28)(3,12,14,30,13,29,31,26)(5,10,20,27,21,8,17,25)(7,22,23,15,24,16,32,19)(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);(1,3,6,13)(2,7)(4,16,18,22)(5,19,21,15)(8,12)(9,26)(10,29)(11,24)(14,17,31,20)(23,25)(27,32)(28,30)1358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 145);