Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7449 |
126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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)(54,56,61,74)(55,62,68,90)(57,76,80,114)(58,72,82,109)(59,71,83,108)(60,85,81,73)(63,92,97,100)(64,87,66,98)(65,86,93,88)(67,99,84,89)(69,94,103,95)(70,104,107,78)(75,110,112,102)(77,106,115,116)(79,105,91,113)(96,117,111,101);(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)(54,55,58,64,81,67,83,93,61,68,82,66,60,84,59,65)(56,69,71,100,85,102,109,117,74,103,108,92,73,110,72,101)(57,75,77,63,91,94,104,96,80,112,115,97,79,95,78,111)(62,76,86,70,89,105,98,116,90,114,88,107,99,113,87,106)Polycyclic presentation:
2^8,53|a^2=d,b^2=h,c^2=fgh,d^2=efgh,e^2=gh,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=defg,e^b=eg,f^b=fgh,g^b=gh,i^b=i^521181 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57215);