Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C97 · Schur multiplier (H2) = C4 |
| direct product | C97×#1410 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C97 | #353 |
| commutator | C42 | C2×C4×C97 |
| Frattini, cosocle | C4⋊Q8 | C22×C97 |
| prime radical | #127×C97 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C97 ◁ C22×C97 ◁ C2×C4×C97 ◁ C4⋊Q8×C97 ◁ G | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 1552 |
A minimal faithful permutation representation — 2 generators on 113 points:
(1,2,3,8,5,9,12,16)(4,6,10,14,13,7,11,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,6)(3,10)(4,13)(7,9)(11,12)(14,15)1649 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 142);