Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Sylow | C53 ⋊ #34336 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D53 |
| commutator | C22×C53 | C24×C4 |
| Fitting | C42×Q8×C53 | C2 |
| socle, prime radical, cosocle | C23×C53 | C25 |
| hypercenter | C42×Q8 | D53 |
| hypocenter | C53 | #34336 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 5 generators on 81 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,57,55)(67,72)(71,74)(73,80)(79,81);(66,69,70,78)(67,73,72,80)(68,75,76,77)(71,79,74,81);(58,60,61,65)(59,62,63,64)(67,72)(71,74)(73,80)(79,81);(66,68,70,76)(67,71,72,74)(69,77,78,75)(73,81,80,79);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(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)(54,55,57,56)(58,59,61,63)(60,64,65,62)(66,67,70,72)(68,74,76,71)(69,73,78,80)(75,81,77,79)Polycyclic presentation:
2^8,53|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=gh,b^a=bf,c^a=cg,e^a=ef,i^a=i^52,d^b=df2168 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 332127);