Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#1425 ⋊ C2 |
| Sylow | C53 ⋊ #13656 |
184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 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)(54,57,59,76,61,77,83,60)(55,63,65,84,68,66,67,92)(56,70,72,103,74,104,73,85)(58,75,80,108,82,109,115,81)(62,87,88,89,90,97,98,116)(64,91,94,111,96,78,95,110)(69,99,100,93,101,102,107,117)(71,79,86,112,106,114,105,113);(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)(57,70)(58,79)(59,72)(60,85)(61,74)(63,87)(64,93)(65,88)(66,97)(67,98)(68,90)(69,78)(71,81)(73,83)(75,86)(76,103)(77,104)(80,112)(82,114)(84,89)(91,101)(92,116)(94,102)(95,99)(96,117)(100,110)(105,109)(106,108)(107,111)(113,115);(54,55,59,65,61,68,83,67)(56,69,72,100,74,101,73,107)(57,66,76,92,77,63,60,84)(58,78,80,110,82,91,115,111)(62,86,88,106,90,105,98,71)(64,75,94,108,96,109,95,81)(70,102,103,117,104,99,85,93)(79,97,112,116,114,87,113,89)Polycyclic presentation:
2^8,53|a^2=e,c^2=e,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,i^b=i^52,i^c=i^521040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90180);