Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#1414 ⋊ C2 |
| Sylow | C53 ⋊ #13608 |
308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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)(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)(54,57,60,73)(55,62,65,88)(56,67,70,97)(58,75,76,71)(59,72,80,106)(61,81,84,85)(63,90,91,86)(64,87,78,104)(66,92,89,111)(68,98,101,95)(69,96,79,105)(74,107,108,102)(77,109,110,103)(82,112,83,113)(93,114,94,115)(99,117,100,116);(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)(55,61,65,84)(57,67,73,97)(58,68,76,101)(59,79,80,69)(62,85,88,81)(63,89,91,66)(64,82,78,83)(71,95,75,98)(72,105,106,96)(74,94,108,93)(77,99,110,100)(86,111,90,92)(87,113,104,112)(102,115,107,114)(103,116,109,117);(54,55)(56,66,69,93)(57,62)(58,74,76,108)(59,78,80,64)(60,65)(61,68,82,99)(63,77)(67,92,96,114)(70,89,79,94)(71,102,75,107)(72,104,106,87)(73,88)(81,98,112,117)(83,100,84,101)(85,95,113,116)(86,103)(90,109)(91,110)(97,111,105,115)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=cg,d^b=dg,e^b=efgh,f^b=fh,i^b=i^52,d^c=dh,i^c=i^52,e^d=egh,f^e=fh467 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89968);