Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#1315 ⋊ C2 |
| Sylow | C53 ⋊ #7347 |
318 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 39 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 — 2 generators on 117 points:
(1,3)(2,9)(4,20)(5,24)(6,29)(7,17)(8,18)(10,38)(11,42)(12,35)(13,36)(14,41)(15,23)(16,26)(19,47)(21,32)(22,49)(25,58)(27,51)(28,34)(30,50)(31,52)(33,43)(37,48)(39,63)(40,59)(44,61)(45,62)(46,53)(54,55)(56,64)(57,60)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116);(1,2,5,11,7,12,26,44,8,13,27,45,6,28,31,43)(3,14,16,46,17,47,51,63,18,38,52,60,29,59,24,56)(4,19,21,53,22,10,37,39,23,40,25,57,30,41,55,64)(9,32,33,15,35,48,42,50,36,58,61,20,34,54,62,49)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)Polycyclic presentation:
2^8,53|a^2=d,c^2=h,d^2=f,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,i^a=i^52,c^b=ch,d^b=de,e^b=eh,g^b=gh,i^b=i^52,d^c=dgh,e^c=eh,e^d=eh1022 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 56937);