Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#1812 ⋊ C2 |
| Sylow | C53 ⋊ #12742 |
878 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 878 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)(55,62)(56,67)(58,72)(59,74)(60,73)(61,82)(63,85)(64,71)(65,86)(66,91)(68,94)(69,96)(70,95)(75,109)(76,104)(77,105)(78,107)(79,106)(80,102)(81,108)(83,100)(84,111)(87,112)(88,101)(89,113)(90,103)(92,93)(97,115)(98,116)(99,114)(110,117);(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)(54,56,60,70)(55,61,64,83)(57,67,73,95)(58,76,79,110)(59,81,77,69)(62,82,71,100)(63,87,88,99)(65,89,80,84)(66,90,92,75)(68,97,98,78)(72,104,106,117)(74,108,105,96)(85,112,101,114)(86,113,102,111)(91,103,93,109)(94,115,116,107);(54,55)(56,66)(57,71,73,62)(58,75)(59,80)(60,64)(61,68)(63,78)(65,77)(67,93,95,91)(69,99)(70,92)(72,103,106,109)(74,86,105,102)(76,89)(79,90)(81,87)(82,94,100,116)(83,98)(84,110)(85,107,101,115)(88,97)(96,112,108,114)(104,111,117,113)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,d^2=gh,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, 85258);