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 ⋊ #13655 |
186 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 186 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)(54,57,59,75,61,76,82,60)(55,63,65,90,67,91,66,83)(56,69,71,100,73,101,72,84)(58,74,79,105,81,106,113,80)(62,86,87,114,89,115,96,88)(64,77,93,107,95,109,94,108)(68,97,98,116,99,117,104,92)(70,78,85,110,103,112,102,111);(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)(57,69)(58,78)(59,71)(60,84)(61,73)(63,86)(64,92)(65,87)(66,96)(67,89)(68,77)(70,80)(72,82)(74,85)(75,100)(76,101)(79,110)(81,112)(83,88)(90,114)(91,115)(93,97)(94,117)(95,116)(98,107)(99,109)(102,106)(103,105)(104,108)(111,113);(54,55,59,65,61,67,82,66)(56,68,71,98,73,99,72,104)(57,63,75,90,76,91,60,83)(58,77,79,107,81,109,113,108)(62,85,87,103,89,102,96,70)(64,74,93,105,95,106,94,80)(69,97,100,116,101,117,84,92)(78,86,110,114,112,115,111,88)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,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^521064 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90175);