Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#1813 ⋊ C2 |
| Sylow | C53 ⋊ #12746 |
886 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 886 conjugacy classes of subgroups — too many to draw. Only its 135 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)(55,63)(56,68)(58,78)(59,82)(60,74)(61,75)(62,85)(64,93)(65,72)(66,91)(67,95)(69,102)(70,99)(71,101)(73,81)(76,105)(77,109)(79,107)(80,112)(83,90)(84,100)(86,115)(87,114)(88,104)(89,94)(92,117)(96,116)(97,113)(98,103)(106,110)(108,111);(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)(55,62)(57,68)(58,77)(59,71)(60,84)(61,70)(63,85)(64,92)(65,88)(66,86)(67,76)(69,80)(72,104)(73,106)(74,100)(75,99)(78,109)(79,103)(81,110)(82,101)(83,87)(89,116)(90,114)(91,115)(93,117)(94,96)(95,105)(97,108)(98,107)(102,112)(111,113);(54,55,59,66,61,65,60,83)(56,67,71,96,70,97,84,92)(57,72,74,91,75,63,82,90)(58,76,79,94,81,108,80,64)(62,77,86,103,88,110,87,69)(68,95,100,117,99,113,101,116)(73,105,107,93,78,111,112,89)(85,106,114,98,104,109,115,102)Polycyclic presentation:
2^8,53|a^2=f,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,e^b=eh,g^b=gh,i^b=i^52,d^c=dh,f^c=fh878 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85278);