Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C53×#510 ⋊ C2 |
| Sylow | C53 ⋊ #13324 |
514 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 95 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)(55,63)(56,69)(58,81)(59,85)(60,77)(61,78)(62,87)(64,95)(65,98)(66,74)(67,93)(68,99)(70,106)(71,75)(72,104)(73,105)(76,84)(79,112)(80,113)(82,111)(83,114)(86,103)(88,91)(89,115)(90,116)(92,97)(94,117)(96,108)(100,102)(101,109)(107,110);(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)(55,62)(57,75)(58,80)(59,71)(60,72)(61,73)(63,91)(64,94)(65,88)(66,89)(67,90)(68,79)(69,77)(70,83)(74,87)(76,106)(78,104)(81,103)(82,86)(84,107)(85,105)(92,99)(93,115)(95,109)(96,101)(97,100)(98,116)(102,108)(110,111)(112,117)(113,114);(54,55,61,67)(56,68,73,101)(57,74,78,98)(58,79,84,96)(59,66,60,65)(62,86,90,70)(63,85,93,77)(64,82,97,83)(69,102,105,117)(71,100,72,94)(75,109,104,99)(76,95,81,92)(80,89,107,88)(87,113,116,110)(91,106,115,103)(108,114,112,111)Polycyclic presentation:
2^8,53|a^2=h,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,f^b=fh,g^b=gh,d^c=dh,e^c=eh,f^c=fh,i^c=i^52866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 88390);