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×#33373 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | C22×#99 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C2×Q8 | C24×C53 |
| prime radical | C53×#3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×Q8×C53 ◁ C2×H(5,𝔽2)×C53 ◁ G | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,6,12,7,13,3,10)(4,9,15,8,5,11,14,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69);(1,3,7,6)(2,8,13,16)(4,14,5,15)(9,10,11,12);(1,4,7,5)(2,9,13,11)(3,14,6,15)(8,10,16,12);(1,3,7,6)(2,10,13,12)(4,15,5,14)(8,11,16,9)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26361);