Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #13230 |
436 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 436 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) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 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)(54,57,73,102,60,76,59,75)(55,62,80,101,65,85,64,72)(56,68,93,112,71,96,70,95)(58,74,91,114,67,90,79,105)(61,81,109,116,83,111,82,99)(63,84,107,103,77,106,86,100)(66,87,98,115,89,113,88,92)(69,94,108,117,78,104,97,110);(54,56,67,69,60,71,58,78)(55,61,77,89,65,83,63,66)(57,68,90,94,76,96,74,104)(59,70,91,97,73,93,79,108)(62,81,106,113,85,111,84,87)(64,82,107,98,80,109,86,88)(72,99,103,115,101,116,100,92)(75,95,114,110,102,112,105,117);(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)(56,66,71,89)(57,72)(58,77,67,63)(59,80)(60,65)(61,69)(62,75)(64,73)(68,92,96,115)(70,98,93,88)(74,103,90,100)(76,101)(78,83)(79,86,91,107)(81,110)(82,108)(84,105,106,114)(85,102)(87,112,113,95)(94,99)(97,109)(104,116)(111,117)Polycyclic presentation:
2^8,53|a^2=f,b^2=df,c^2=eh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,i^a=i^52,d^b=dg,f^b=fg,g^b=gh,f^d=fh1178 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 87911);