Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#1382 ⋊ C2 |
| Sylow | C53 ⋊ #12832 |
358 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 111 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)(55,62)(56,67)(58,72)(59,74)(60,73)(61,80)(63,83)(64,71)(65,84)(66,86)(68,90)(69,92)(70,91)(75,104)(76,102)(77,101)(78,99)(79,103)(81,97)(82,108)(85,98)(87,89)(88,110)(93,114)(94,113)(95,115)(96,112)(100,105)(106,116)(107,111)(109,117);(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)(54,56,58,68,59,70,76,94,60,69,77,93,79,95,106,109)(55,61,63,66,65,82,75,88,64,81,85,87,78,107,105,96)(57,67,72,90,74,91,102,113,73,92,101,114,103,115,116,117)(62,80,83,86,84,108,104,110,71,97,98,89,99,111,100,112);(54,55)(56,66,69,87)(57,71,73,62)(58,75)(59,78)(60,64)(61,68,81,93)(63,76)(65,79)(67,89)(70,96,95,88)(72,100,101,104)(74,84,103,99)(77,105)(80,90)(82,109,107,94)(83,102,98,116)(85,106)(86,92)(91,110)(97,114)(108,117)(111,113)(112,115)Polycyclic presentation:
2^8,53|a^2=f,b^2=d,d^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,f^b=fh,i^b=i^52,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, 85723);