Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #13653 |
188 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 188 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) = 121 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 121 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)(54,56,57,55)(59,66)(65,87)(67,90)(68,91)(69,85)(70,97)(75,96)(81,115)(88,117)(89,95)(92,118)(93,106)(94,119)(98,105)(102,121)(107,113);(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)(58,60,64,74)(59,65,66,87)(61,76,80,110)(62,82,84,116)(63,86,83,73)(67,92,90,118)(68,94,91,119)(69,95,85,89)(70,96,97,75)(71,99,101,78)(72,103,104,77)(79,114,112,109)(81,88,115,117)(93,121,106,102)(98,107,105,113)(100,111,108,120);(54,55,57,56)(58,59)(60,70)(61,75)(62,81)(63,85)(64,66)(65,71)(67,78)(68,77)(69,83)(72,102)(73,105)(74,97)(76,94)(79,113)(80,96)(82,92)(84,115)(86,98)(87,101)(88,109)(89,108)(90,99)(91,103)(93,120)(95,100)(104,121)(106,111)(107,112)(110,119)(114,117)(116,118)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=e,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,i^c=i^521040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90163);