Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #13666 |
288 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 288 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) = 6784 |
A minimal faithful permutation representation — 3 generators on 121 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)(55,57)(58,61,64,79)(59,66,70,69)(60,72,76,75)(62,77,83,112)(63,78,86,97)(65,88,90,100)(67,91,96,95)(68,92,99,84)(71,101,102,108)(73,103,87,105)(74,104,107,85)(80,109,114,113)(81,110,116,115)(82,111,93,118)(89,98,117,121)(94,106,119,120);(58,60)(59,65)(61,72)(62,81)(63,85)(64,76)(66,88)(67,94)(68,98)(69,100)(70,90)(71,80)(73,82)(74,78)(75,79)(77,110)(83,116)(84,89)(86,104)(87,93)(91,106)(92,117)(95,120)(96,119)(97,107)(99,121)(101,109)(102,114)(103,111)(105,118)(108,113)(112,115);(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)(56,57)(58,59,64,70)(60,71,76,102)(61,66,79,69)(62,80,83,114)(63,84,86,92)(65,87,90,73)(67,93,96,82)(68,97,99,78)(72,101,75,108)(74,106,107,120)(77,109,112,113)(81,98,116,121)(85,94,104,119)(88,105,100,103)(89,110,117,115)(91,118,95,111)Polycyclic presentation:
2^8,53|a^2=h,c^2=h,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,i^a=i^52,d^b=df,f^b=fg,g^b=gh1193 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90235);