Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#1358 ⋊ C2 |
| Sylow | C53 ⋊ #11990 |
268 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(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)(54,57)(55,62,66,87)(56,68)(58,74)(59,75)(60,73)(61,81,84,108)(63,88,65,72)(64,86,92,113)(67,94,83,106)(69,97)(70,98)(71,96)(76,104)(77,85)(78,102)(79,105)(80,103)(82,107,109,117)(89,114,91,101)(90,112,95,116)(93,111)(99,115)(100,110);(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)(54,56,60,71)(55,61,66,84)(57,68,73,96)(58,76,78,99)(59,79,80,70)(62,81,87,108)(63,83,65,67)(64,90,92,95)(69,77,100,93)(72,94,88,106)(74,104,102,115)(75,105,103,98)(82,89,109,91)(85,110,111,97)(86,112,113,116)(101,107,114,117);(54,55,58,64,59,63,77,89,60,66,78,92,80,65,93,91)(56,67,69,95,70,84,99,109,71,83,100,90,79,61,76,82)(57,72,74,101,75,62,85,86,73,88,102,114,103,87,111,113)(68,81,97,107,98,94,115,116,96,108,110,117,105,106,104,112)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=g,d^2=g,e^2=fg,f^2=h,b^a=bd,c^a=cf,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,d^c=dh,i^c=i^52632 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 81087);