Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#8490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #379 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#538 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2,5,10,6,9,20,24)(3,11,14,25,13,7,18,22)(4,15,17,26,16,8,19,23)(12,27,29,32,28,21,30,31)(33,34)(35,43)(36,47,49,40)(37,50)(38,42)(39,45)(41,52)(44,58,60,57)(46,56)(48,55,64,62)(51,54)(53,59,61,63)(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);(1,3,6,13)(2,7,9,11)(4,12,16,28)(5,18,20,14)(8,21,15,27)(10,25,24,22)(17,30,19,29)(23,32,26,31)(33,35,38,46)(34,39,42,54)(36,44,49,60)(37,51,52,45)(40,53,47,61)(41,56,50,43)(48,63,64,59)(55,58,62,57);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,15)(10,26)(11,27)(13,28)(14,30)(17,20)(18,29)(22,32)(23,24)(25,31)(33,36)(34,40)(35,44)(37,48)(38,49)(39,53)(41,55)(42,47)(43,57)(45,59)(46,60)(50,62)(51,63)(52,64)(54,61)(56,58)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1478);