Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #30238 |
436 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 135 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)(54,58,61,80,60,79,59,78)(55,64,81,105,66,93,65,92)(56,69,82,106,71,100,70,99)(57,73,76,103,75,102,74,77)(62,84,85,111,86,112,67,95)(63,88,94,115,90,114,89,91)(68,96,101,116,97,104,72,98)(83,107,87,113,109,117,108,110);(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)(54,57,60,75)(55,63,66,90)(56,68,71,97)(58,77,79,103)(59,76,61,74)(62,83,86,109)(64,91,93,115)(65,94,81,89)(67,87,85,108)(69,98,100,116)(70,101,82,72)(73,80,102,78)(84,110,112,113)(88,105,114,92)(95,107,111,117)(96,106,104,99);(54,56,60,71)(55,62)(57,72,75,101)(58,69,79,100)(59,70,61,82)(63,87)(64,84)(65,67)(66,86)(68,74,97,76)(73,98,102,116)(77,104,103,96)(78,99,80,106)(81,85)(83,89)(88,113)(90,108)(91,107)(92,95)(93,112)(94,109)(105,111)(110,114)(115,117);(54,55,59,65,60,66,61,81)(56,67,82,62,71,85,70,86)(57,63,74,89,75,90,76,94)(58,64,78,92,79,93,80,105)(68,87,101,109,97,108,72,83)(69,95,106,84,100,111,99,112)(73,88,77,91,102,114,103,115)(96,113,116,117,104,110,98,107)Polycyclic presentation:
2^8,53|a^2=fg,b^2=f,c^2=h,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,g^b=gh,d^c=de,e^c=eh,g^c=gh,i^c=i^521178 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 281717);