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×#25197 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ G | |
968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,10,7,11,5,20)(3,12,14,8,15,24,21,25)(4,16,18,28,19,27,22,9)(13,26,31,32,17,23,30,29)(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);(1,3,6,14,7,15,5,21)(2,8,10,24,11,25,20,12)(4,17,18,30,19,13,22,31)(9,26,16,32,28,23,27,29);(1,4)(2,9)(3,13)(5,18)(6,22)(7,19)(8,23)(10,27)(11,28)(12,29)(14,30)(15,17)(16,20)(21,31)(24,32)(25,26);(1,2,7,11)(3,8,15,25)(4,9,19,28)(5,20,6,10)(12,14,24,21)(13,23,17,26)(16,22,27,18)(29,30,32,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 18185);