Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #7479 |
162 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 91 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)(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)(54,56,59,72,60,73,82,86,81,106,114,115,113,111,79,70,61,74,83,104,85,109,78,112,58,71,80,105,77,103,57,76)(55,62,65,88,66,89,94,101,64,87,96,116,97,107,63,92,68,91,75,110,99,117,93,102,67,90,98,108,84,100,95,69);(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)(54,55,58,64,61,68,81,67)(56,69,71,101,74,92,106,102)(57,75,78,98,79,65,82,96)(59,63,80,93,83,95,114,94)(60,84,77,66,85,97,113,99)(62,86,87,76,91,112,90,70)(72,107,105,117,104,100,115,89)(73,108,103,88,109,116,111,110)Polycyclic presentation:
2^8,53|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=f,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fg,g^a=gh,i^a=i^52,c^b=ch,i^b=i^521196 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57290);