Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #7228 |
252 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(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)(54,56,59,71,75,99,77,101,61,73,57,69,60,72,82,104)(55,62,65,85,66,86,81,108,67,87,63,68,90,97,74,98)(58,70,78,83,106,109,95,115,80,103,76,100,79,102,88,110)(64,91,92,96,93,114,107,116,94,84,89,111,113,112,105,117);(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)(54,55,58,64)(56,68,70,96)(57,74,76,105)(59,81,78,107)(60,66,79,93)(61,67,80,94)(62,83,84,71)(63,88,89,82)(65,95,92,77)(69,87,100,91)(72,98,102,116)(73,85,103,111)(75,90,106,113)(86,110,112,104)(97,115,114,101)(99,108,109,117)Polycyclic presentation:
2^8,53|a^2=d,b^2=e,c^2=fgh,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,d^b=dg,i^b=i^52,d^c=dh872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 56637);