Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#22257 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C22×D4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C22×C53 | C2×C4×D4 |
| prime radical | C23×C53 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 101 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,14)(15,16)(17,18,22,26,23,27,21,36)(19,24,30,38,31,39,29,45)(20,32,35,40,33,46,34,25)(28,41,44,47,42,48,43,37)(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);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,22,30,23,31,21,29)(18,24,26,38,27,39,36,45)(20,28,35,44,33,42,34,43)(25,37,32,41,40,47,46,48);(2,7)(3,10)(8,12)(11,16)(17,20,21,34,23,33,22,35)(18,25,26,32,27,40,36,46)(19,28,29,43,31,42,30,44)(24,37,38,41,39,47,45,48);(1,4,5,13)(2,8,7,12)(3,11,10,16)(6,14,9,15)(20,33)(25,40)(28,42)(32,46)(34,35)(37,47)(41,48)(43,44)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15245);