Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #33687 |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 125 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)(54,56,57,60)(55,59,58,61)(64,76)(69,74)(75,104)(77,107)(78,108)(90,100)(91,101)(92,102)(103,113)(105,122)(106,112)(109,118)(110,125)(111,123)(119,121)(120,124);(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)(62,65,66,80)(63,70,71,94)(64,75,76,104)(67,86,83,117)(68,89,84,82)(69,90,74,100)(72,98,96,116)(73,99,88,95)(77,110,107,125)(78,112,108,106)(79,114,115,85)(81,97,93,87)(91,120,101,124)(92,121,102,119)(103,118,113,109)(105,123,122,111);(55,58)(59,61)(62,64)(63,69)(65,75)(66,76)(67,77)(68,78)(70,90)(71,74)(72,91)(73,92)(79,113)(80,104)(81,105)(82,106)(83,107)(84,108)(85,118)(86,110)(87,111)(88,102)(89,112)(93,122)(94,100)(95,119)(96,101)(97,123)(98,120)(99,121)(103,115)(109,114)(116,124)(117,125);(54,55)(56,59)(57,58)(60,61)(62,63,66,71)(64,74,76,69)(65,79,80,115)(67,85,83,114)(68,88,84,73)(70,93,94,81)(72,97,96,87)(75,103,104,113)(77,109,107,118)(78,92,108,102)(82,116,89,98)(86,99,117,95)(90,105,100,122)(91,111,101,123)(106,120,112,124)(110,119,125,121)Polycyclic presentation:
2^8,53|a^2=h,c^2=h,d^2=eh,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^c=fg,g^c=gh,i^d=i^52800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 324786);