Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #33414 |
622 conjugacy classes of subgroups; only the 391 normal subgroups are shown.
This group has 622 conjugacy classes of subgroups — too many to draw. Only its 391 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 — 4 generators on 117 points:
(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)(57,74)(63,86)(69,96)(75,102)(76,77)(83,108)(87,110)(88,105)(90,111)(95,104)(97,115)(98,99)(103,117)(107,114)(109,112)(113,116);(54,57)(55,63)(56,69)(58,75)(59,76)(60,77)(61,74)(62,83)(64,87)(65,88)(66,86)(67,90)(68,95)(70,97)(71,98)(72,99)(73,96)(78,103)(79,104)(80,102)(81,105)(82,107)(84,109)(85,108)(89,110)(91,112)(92,113)(93,111)(94,114)(100,115)(101,116)(106,117);(54,56,68,94,60,72,80,100,61,73,79,82,59,71,58,70)(55,62,78,92,65,84,89,93,66,85,106,101,81,91,64,67)(57,69,95,114,77,99,102,115,74,96,104,107,76,98,75,97)(63,83,103,113,88,109,110,111,86,108,117,116,105,112,87,90);(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)(54,55,59,65,61,66,60,81)(56,67,71,92,73,93,72,101)(57,63,76,88,74,86,77,105)(58,78,79,89,80,106,68,64)(62,82,84,100,85,94,91,70)(69,90,98,113,96,111,99,116)(75,103,104,110,102,117,95,87)(83,107,109,115,108,114,112,97)Polycyclic presentation:
2^8,53|a^2=f,b^2=eg,e^2=gh,f^2=h,g^2=h,b^a=be,e^a=eg,g^a=gh,i^a=i^52,d^c=dh2144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 322628);