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 ⋊ #12744 |
874 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 874 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)(54,57,61,75)(55,63,65,72)(56,68,71,100)(58,73,80,107)(59,81,60,74)(62,84,85,102)(64,88,92,103)(66,90,82,89)(67,93,94,96)(69,97,101,117)(70,99,83,98)(76,108,109,104)(77,105,110,116)(78,111,79,106)(86,113,87,112)(91,114,95,115);(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)(54,56)(55,62)(57,68)(58,77)(59,70)(60,83)(61,71)(63,84)(64,91)(65,85)(66,86)(67,76)(69,79)(72,102)(73,105)(74,98)(75,100)(78,101)(80,110)(81,99)(82,87)(88,114)(89,112)(90,113)(92,95)(93,108)(94,109)(96,104)(97,106)(103,115)(107,116)(111,117);(54,55,59,66,61,65,60,82)(56,67,70,95,71,94,83,91)(57,72,74,90,75,63,81,89)(58,76,78,92,80,109,79,64)(62,77,86,101,85,110,87,69)(68,96,98,115,100,93,99,114)(73,104,106,103,107,108,111,88)(84,105,112,97,102,116,113,117)Polycyclic presentation:
2^8,53|a^2=f,c^2=h,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,g^b=gh,i^b=i^52,f^c=fh872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85268);