Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #11771 |
292 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 65 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)(54,57,60,76,61,77,59,75)(55,63,83,109,66,90,65,89)(56,68,72,99,70,98,71,73)(58,74,82,108,80,106,81,107)(62,84,93,112,86,111,85,87)(64,88,94,115,92,114,78,105)(67,95,102,117,96,113,91,103)(69,97,79,110,101,116,100,104);(54,56,61,70)(55,62,66,86)(57,73,77,99)(58,79,80,100)(59,72,60,71)(63,87,90,112)(64,91,92,102)(65,93,83,85)(67,94,96,78)(68,76,98,75)(69,81,101,82)(74,97,106,116)(84,109,111,89)(88,113,114,95)(103,115,117,105)(104,107,110,108);(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)(54,55)(56,67,70,96)(57,63)(58,78,80,94)(59,65)(60,83)(61,66)(62,69)(64,81,92,82)(68,95,98,113)(71,91,72,102)(73,103,99,117)(74,105,106,115)(75,89)(76,109)(77,90)(79,93)(84,97)(85,100)(86,101)(87,104)(88,107,114,108)(110,112)(111,116)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=fh,d^2=gh,f^2=h,g^2=h,b^a=bd,d^a=df,g^a=gh,i^a=i^52,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,i^c=i^52,e^d=eh554 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 80116);