Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#13230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1964 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| prime radical | C2×#41×C53 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×#131×C53 ◁ G | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2)(3,13,18,36)(4,19)(5,24,14,10)(6,27)(7,12)(8,16)(9,22)(11,20)(15,39,43,62)(17,45,40,35)(21,50,37,47)(23,48)(25,30)(26,33,38,54)(28,57)(29,55)(31,52,53,61)(32,49)(34,59,60,42)(41,46)(44,56)(51,63)(58,64)(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);(1,3,14,16,7,18,5,25)(2,8,24,36,12,30,10,13)(4,15,37,41,23,43,21,51)(6,17,38,44,20,40,26,55)(9,28,53,60,32,58,31,34)(11,29,54,45,27,56,33,35)(19,46,50,62,48,63,47,39)(22,42,61,57,49,59,52,64);(1,4,20,49,7,23,6,22)(2,9,27,48,12,32,11,19)(3,15,40,59,18,43,17,42)(5,21,38,61,14,37,26,52)(8,28,56,63,30,58,29,46)(10,31,54,50,24,53,33,47)(13,34,45,62,36,60,35,39)(16,41,55,64,25,51,44,57)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6218);