Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13726 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C53 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C53 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ C4×C8×C53 ◁ C4×C16×C53 ◁ G | |
230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 121 points:
(2,4)(5,6,11,16)(7,17,22,43)(8,23,26,55)(9,27,29,57)(10,30,28,15)(12,31,34,19)(13,35,37,18)(14,38,39,24)(20,49,51,68)(21,52,50,42)(25,56,45,54)(32,62,63,67)(33,48,58,61)(36,46,53,64)(40,60,65,44)(41,59,66,47)(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);(5,7,18,44,9,20,45,61,28,50,64,62,38,59,26,31,11,22,35,60,29,51,25,48,10,21,46,67,24,47,8,19)(6,12,23,41,14,32,53,52,30,58,56,68,57,65,37,43,16,34,55,66,39,63,36,42,15,33,54,49,27,40,13,17);(1,2,3,4)3710 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6714);