Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #12437 |
214 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 117 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)(54,57)(55,64)(56,71)(58,78)(59,83)(60,79)(61,84)(62,80)(63,86)(65,77)(66,96)(67,92)(68,97)(69,93)(70,98)(72,102)(73,106)(74,103)(75,107)(76,104)(81,110)(82,91)(85,111)(87,114)(88,112)(89,101)(90,113)(94,109)(95,108)(99,116)(100,115)(105,117);(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)(54,56,59,73,60,74,82,85,62,76,81,105,61,75,58,72)(55,63,66,87,67,88,95,100,69,90,94,99,68,89,65,70)(57,71,83,106,79,103,91,111,80,104,110,117,84,107,78,102)(64,86,96,114,92,112,108,115,93,113,109,116,97,101,77,98);(54,55,62,69)(56,70,76,100)(57,77,80,108)(58,66,82,94)(59,65,81,95)(60,68,61,67)(63,85,90,72)(64,91,93,78)(71,101,104,112)(73,89,105,88)(74,99,75,87)(79,109,84,96)(83,97,110,92)(86,103,113,107)(98,111,115,102)(106,116,117,114)Polycyclic presentation:
2^8,53|a^2=h,b^2=df,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,i^b=i^52,e^c=eg,g^c=gh635 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 83811);