Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#25198 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C53×#3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ G | |
958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,11,7,5,24,10)(3,12,16,14,17,27,15,8)(4,18,22,20,23,28,21,9)(13,25,30,32,19,26,31,29)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,6,16,7,17,24,15)(2,8,11,12,5,14,10,27)(4,19,22,31,23,13,21,30)(9,26,18,29,20,25,28,32);(1,4,7,23)(2,9,5,20)(3,13,17,19)(6,21,24,22)(8,25,14,26)(10,18,11,28)(12,29,27,32)(15,30,16,31);(1,5)(2,7)(3,14)(4,20)(6,10)(8,17)(9,23)(11,24)(12,15)(13,26)(16,27)(18,21)(19,25)(22,28)(29,30)(31,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 18186);