Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#2176 ⋊ C2 |
| Sylow | C53 ⋊ #13657 |
182 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,57,60,74,61,75,84,59)(55,63,66,65,67,82,85,97)(56,69,86,108,71,106,72,83)(58,78,80,79,81,73,113,107)(62,87,98,96,88,89,90,101)(64,93,94,99,95,91,112,76)(68,100,111,117,102,92,103,104)(70,77,114,116,109,105,110,115);(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)(58,77)(59,83)(60,86)(61,71)(63,87)(64,92)(65,96)(66,98)(67,88)(68,99)(70,107)(72,84)(73,110)(74,108)(75,106)(76,102)(78,114)(79,109)(80,116)(81,105)(82,89)(85,90)(91,111)(93,103)(94,104)(95,100)(97,101)(112,117)(113,115);(54,55)(56,68,71,102)(57,65)(58,76)(59,82)(60,85)(61,67)(62,70,88,109)(63,74)(64,79)(66,84)(69,104,106,117)(72,111,86,103)(73,112)(75,97)(77,96,105,101)(78,94)(80,91)(81,99)(83,100,108,92)(87,115,89,116)(90,114,98,110)(93,113)(95,107)Polycyclic presentation:
2^8,53|a^2=e,c^2=eg,d^2=fgh,f^2=gh,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fg,g^a=gh,d^b=dfh,e^b=eh,f^b=fg,g^b=gh,i^b=i^52,d^c=dh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90184);