Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C53×C64 · Schur multiplier (H2) = C22 |
| direct product | C53×C22⋊C64 |
| unambiguous semidirect product | C22⋊(C53×C64) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C53 | C22 |
| commutator | C2 | C2×C53×C64 |
| Frattini, cosocle | C2×C32 | C22×C53 |
| socle | C22×C53 | C2×C32 |
| prime radical | C23×C53 | C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C53 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 2 generators on 121 points:
(1,2)(3,4)(5,6,7,12,8,13,17,27,9,14,18,28,21,31,37,47,10,15,19,29,22,32,38,48,24,34,40,50,43,53,57,62,11,16,20,30,23,33,39,49,25,35,41,51,44,54,58,63,26,36,42,52,45,55,59,64,46,56,60,65,61,66,67,68)(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);(2,4)8480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 500);