Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| Sylow | C53 ⋊ #7536 |
138 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 2 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)(54,56)(55,60,64,75)(57,70)(58,73)(59,68)(61,78,80,81)(62,77,63,76)(65,79,82,69)(66,71)(67,74)(72,84)(83,85);(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)(54,55)(56,65,67,81)(57,69,71,61)(58,63,74,62)(59,64)(60,66,75,83)(68,82,73,78)(70,76)(72,79,85,80)(77,84)Polycyclic presentation:
2^8,53|a^2=d,b^2=gh,c^2=fgh,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cfgh,d^b=defh,e^b=egh,f^b=fh,i^b=i^52,d^c=dfg,e^d=egh,f^d=fh467 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57442);