Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7425 |
136 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 43 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 — 2 generators on 117 points:
(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)(54,56,62,74)(55,63,67,93)(57,76,81,104)(58,73,61,70)(59,85,60,87)(64,95,98,116)(65,90,88,92)(66,99,100,117)(68,101,103,75)(69,79,105,80)(71,77,72,83)(78,114,82,112)(84,115,86,91)(89,94,106,113)(96,107,97,108)(102,111,109,110);(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)(54,55,58,65,62,67,61,88)(56,68,70,102,74,103,73,109)(57,75,78,110,81,101,82,111)(59,84,83,100,60,86,77,66)(63,89,90,105,93,106,92,69)(64,94,96,79,98,113,97,80)(71,107,85,116,72,108,87,95)(76,99,112,115,104,117,114,91)Polycyclic presentation:
2^8,53|a^2=d,b^2=h,c^2=fgh,d^2=h,e^2=g,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,i^a=i^52,c^b=cf,d^b=defgh,e^b=egh,f^b=fgh,g^b=gh,d^c=dh707 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57144);