Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#33371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | C22×#99 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C2×Q8 | C24×C53 |
| prime radical | C53×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×Q8×C53 ◁ C2×H(5,𝔽2)×C53 ◁ G | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,7,12,8,13,5,11)(3,14,16,30,17,24,22,32)(4,10,18,28,6,20,19,29)(9,23,26,21,27,31,25,15)(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)(2,9)(4,15)(5,16)(6,21)(7,22)(8,17)(10,24)(11,25)(12,26)(13,27)(14,20)(18,31)(19,23)(28,32)(29,30);(1,4,8,6)(2,10,13,20)(3,15,17,21)(5,19,7,18)(9,24,27,14)(11,29,12,28)(16,23,22,31)(25,30,26,32);(1,5,8,7)(2,11,13,12)(3,16,17,22)(4,18,6,19)(9,25,27,26)(10,28,20,29)(14,30,24,32)(15,31,21,23)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26359);