Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #12827 |
376 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(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)(54,57,60,75)(55,62,66,89)(56,68,71,99)(58,76,78,73)(59,81,82,110)(61,83,85,115)(63,88,65,87)(64,92,93,105)(67,94,86,113)(69,100,102,97)(70,103,80,112)(72,90,106,79)(74,108,109,77)(84,96,91,117)(95,104,116,114)(98,111,107,101);(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)(54,56,60,71)(55,61)(57,68,75,99)(58,69,78,102)(59,80,82,70)(62,83)(63,67)(64,91)(65,86)(66,85)(72,104)(73,97,76,100)(74,107,109,98)(77,111,108,101)(79,95)(81,112,110,103)(84,93)(87,113)(88,94)(89,115)(90,116)(92,117)(96,105)(106,114);(54,55)(56,67,71,86)(57,72)(58,63)(59,79)(60,66)(61,69,85,102)(62,74)(64,77)(65,78)(68,96,99,117)(70,84,80,91)(73,105)(75,106)(76,92)(81,87)(82,90)(83,112,115,103)(88,110)(89,109)(93,108)(94,107,113,98)(95,111,116,101)(97,114,100,104)Polycyclic presentation:
2^8,53|a^2=f,b^2=dg,c^2=h,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh,i^c=i^52872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85688);