Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#2221 ⋊ C2 |
| Sylow | C53 ⋊ #13669 |
288 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 288 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)(54,56)(55,62)(57,68)(58,71)(59,70)(60,72)(61,84)(63,89)(64,91)(65,90)(66,92)(67,79)(69,80)(73,87)(74,77)(75,82)(76,99)(78,100)(81,114)(83,102)(85,105)(86,95)(88,103)(93,113)(94,115)(96,104)(97,116)(98,111)(101,110)(106,117)(107,109)(108,112);(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)(55,61)(57,74)(58,69)(59,82)(60,72)(62,84)(63,93)(64,85)(65,96)(66,88)(67,73)(68,77)(70,75)(71,80)(76,106)(78,108)(79,87)(81,86)(83,101)(89,113)(90,104)(91,105)(92,103)(94,116)(95,114)(97,115)(98,109)(99,117)(100,112)(102,110)(107,111);(54,55)(56,67)(57,73)(58,79)(59,81)(60,66)(61,68)(62,69)(63,77)(64,80)(65,75)(70,90)(71,92)(72,91)(74,96)(76,84)(78,105)(82,93)(83,113)(85,99)(86,107)(87,98)(88,100)(89,106)(94,117)(95,112)(97,110)(101,114)(102,116)(103,111)(104,109)(108,115)Polycyclic presentation:
2^8,53|d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,i^b=i^52,e^c=eh713 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90250);