Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #12902 |
278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(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)(54,57)(55,62)(56,66,67,89)(58,72)(59,73)(60,71)(61,77,78,88)(63,83)(64,84)(65,82)(68,90,91,101)(69,70,92,95)(74,99)(75,97)(76,98)(79,102,103,112)(80,81,104,94)(85,110)(86,108)(87,109)(93,96,113,106)(100,115)(105,107,116,114)(111,117);(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)(55,61)(57,70,71,95)(58,68)(59,69)(60,67)(62,81,82,94)(63,79)(64,80)(65,78)(66,73,89,98)(72,96,97,106)(74,93)(75,91)(76,92)(77,84,88,109)(83,107,108,114)(85,105)(86,103)(87,104)(90,99,101,115)(100,113)(102,110,112,117)(111,116);(54,55,58,63,59,64,74,85,60,65,75,86,76,87,100,111)(56,61,68,79,69,80,93,105,67,78,91,103,92,104,113,116)(57,62,72,83,73,84,99,110,71,82,97,108,98,109,115,117)(66,88,90,112,70,94,96,114,89,77,101,102,95,81,106,107)Polycyclic presentation:
2^8,53|a^2=f,b^2=d,c^2=e,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=cg,e^b=eh,d^c=dh,i^c=i^521508 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85989);