Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #13390 |
382 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 93 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)(54,57,58,74,60,76,80,77)(55,62,63,90,65,91,78,92)(56,67,68,99,71,73,70,105)(59,82,79,88,83,111,93,75)(61,84,66,96,87,89,86,117)(64,94,81,107,95,108,112,72)(69,102,101,100,104,114,103,110)(85,109,97,115,116,98,113,106);(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)(55,61)(57,73)(58,70)(59,69)(60,71)(62,89)(63,86)(64,85)(65,87)(66,78)(67,76)(68,80)(72,106)(74,99)(75,110)(77,105)(79,103)(81,113)(82,114)(83,104)(84,91)(88,100)(90,96)(92,117)(93,101)(94,98)(95,116)(97,112)(102,111)(107,115)(108,109);(54,55,60,65)(56,66,71,86)(57,72,76,107)(58,78,80,63)(59,81,83,112)(61,70,87,68)(62,88,91,75)(64,93,95,79)(67,98,73,109)(69,85,104,116)(74,108,77,94)(82,92,111,90)(84,102,89,114)(96,110,117,100)(97,101,113,103)(99,115,105,106)Polycyclic presentation:
2^8,53|a^2=h,c^2=d,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,c^b=cf,d^b=dh,e^c=egh,f^c=fh,g^c=gh,i^c=i^52866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 88790);