Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #12878 |
362 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 362 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)(54,57,60,84,61,73,59,75)(55,63,83,90,64,91,66,89)(56,68,70,99,71,97,82,100)(58,78,80,74,81,112,79,106)(62,85,87,114,67,95,88,116)(65,94,108,105,92,104,76,72)(69,102,103,98,77,110,111,107)(86,115,101,96,93,117,109,113);(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)(55,62)(57,68)(58,77)(59,82)(60,70)(61,71)(63,85)(64,67)(65,93)(66,88)(69,81)(72,96)(73,97)(74,107)(75,100)(76,101)(78,110)(79,103)(80,111)(83,87)(84,99)(86,92)(89,116)(90,114)(91,95)(94,117)(98,106)(102,112)(104,115)(105,113)(108,109);(54,55,59,66,61,64,60,83)(56,67,70,88,71,62,82,87)(57,72,75,104,73,105,84,94)(58,76,79,92,81,108,80,65)(63,78,89,106,91,112,90,74)(68,96,99,117,97,113,100,115)(69,101,103,93,77,109,111,86)(85,102,114,98,95,110,116,107)Polycyclic presentation:
2^8,53|a^2=f,c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh,i^c=i^52878 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85908);