Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #17307 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D53 |
| commutator, socle, prime radical, cosocle | C24×C53 | C24 |
| Fitting | C53×#1444 | C2 |
| hypercenter | #1444 | D53 |
| hypocenter | C53 | #17307 |
| LCS | G ▷ C24×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ #1444 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 85 points:
(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)(56,60)(58,61)(63,67)(66,69)(71,75)(74,77)(78,80,81,85)(79,83,82,84);(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)(55,59)(58,61)(70,72,73,76)(71,74,75,77)(80,85)(83,84);(54,56,57,60)(55,58,59,61)(62,64,65,68)(63,66,67,69)(72,76)(74,77)(79,82)(80,85);(54,55,57,59)(56,58,60,61)(62,63)(64,66)(65,67)(68,69)(70,71)(72,74)(73,75)(76,77)(78,79,81,82)(80,84,85,83)Polycyclic presentation:
2^8,53|a^2=ef,b^2=efgh,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de,i^d=i^521190 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 135126);