Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #12885 |
234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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)(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)(54,57,58,76,61,79,60,77)(55,63,64,92,67,94,66,74)(56,69,70,75,73,103,71,105)(59,82,80,110,84,78,83,90)(62,86,68,91,89,101,87,108)(65,97,95,107,99,93,98,113)(72,106,104,111,81,102,85,109)(88,115,100,117,96,114,112,116);(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)(54,56,58,71,61,73,60,70)(55,62,64,87,67,89,66,68)(57,75,77,103,79,105,76,69)(59,81,80,104,84,72,83,85)(63,91,74,101,94,108,92,86)(65,96,95,100,99,88,98,112)(78,111,110,102,82,109,90,106)(93,117,107,114,97,116,113,115);(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)(54,55,59,65,60,66,83,98,61,67,84,99,58,64,80,95)(56,68,72,100,70,89,104,96,73,87,81,112,71,62,85,88)(57,74,78,107,77,94,110,97,79,92,82,113,76,63,90,93)(69,101,102,115,75,108,109,117,103,86,106,114,105,91,111,116)Polycyclic presentation:
2^8,53|a^2=f,b^2=e,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,i^a=i^52,c^b=cg,d^b=dh,f^b=fh,i^b=i^52,e^c=eh,f^c=fh,i^c=i^521196 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85937);