Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×Q32×C53 ⋊ C2 |
| Sylow | C53 ⋊ #13322 |
520 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 6784 |
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)(55,57)(58,61)(59,67)(60,73)(62,84)(63,88)(64,80)(65,81)(66,90)(68,97)(69,100)(70,78)(71,95)(72,101)(74,108)(75,110)(76,106)(77,107)(79,87)(82,115)(83,116)(85,114)(86,117)(89,105)(91,111)(92,119)(93,120)(94,99)(96,104)(98,112)(102,121)(103,118)(109,113);(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)(54,55)(56,57)(58,60,65,77)(59,66,71,93)(61,73,81,107)(62,83,87,109)(63,76,64,75)(67,90,95,120)(68,96,99,102)(69,92,70,91)(72,98,103,82)(74,85,89,86)(78,111,100,119)(79,113,84,116)(80,110,88,106)(94,121,97,104)(101,112,118,115)(105,117,108,114);(58,59,65,71)(60,72,77,103)(61,78,81,100)(62,82,87,98)(63,70,64,69)(66,89,93,74)(67,88,95,80)(68,85,99,86)(73,104,107,121)(75,96,76,102)(79,97,84,94)(83,91,109,92)(90,113,120,116)(101,110,118,106)(105,111,108,119)(112,117,115,114)Polycyclic presentation:
2^8,53|a^2=h,b^2=h,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,g^b=gh,d^c=dh,e^c=eh,i^c=i^521184 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 88376);