Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ C4×Q64 |
190 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 121 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)(54,56,57,55);(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)(58,60,62,72)(59,65,67,88)(61,76,78,108)(63,84,81,116)(64,86,82,74)(66,92,93,119)(68,97,95,121)(69,98,85,90)(70,99,100,75)(71,91,87,79)(73,96,103,77)(80,114,112,110)(83,89,115,117)(94,120,106,104)(101,107,105,113)(102,111,109,118);(54,55,57,56)(58,59,62,67)(60,70,72,100)(61,75,78,99)(63,83,81,115)(64,85,82,69)(65,87,88,71)(66,91,93,79)(68,96,95,77)(73,104,103,120)(74,105,86,101)(76,97,108,121)(80,113,112,107)(84,92,116,119)(89,114,117,110)(90,102,98,109)(94,111,106,118)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=eh,d^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,i^c=i^521064 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90151);