Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #33693 |
512 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(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)(56,67)(57,71,72,74)(58,78)(61,83)(62,87,88,89)(63,93)(66,95,96,103)(69,98)(70,92,105,108)(73,77,86,106)(75,110)(80,90)(82,113,114,116)(85,102)(94,117,104,115)(97,111,107,100);(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)(54,57)(55,62)(56,66)(58,77)(59,71)(60,72)(61,82)(63,92)(64,87)(65,88)(67,96)(68,100)(69,103)(70,75)(73,80)(74,79)(76,111)(78,106)(81,89)(83,114)(84,115)(85,116)(86,90)(91,117)(93,108)(94,99)(95,98)(97,101)(102,113)(104,109)(105,110)(107,112);(54,56,60,67)(55,61,65,83)(57,66,72,96)(58,76,78,68)(59,69,79,98)(62,82,88,114)(63,91,93,84)(64,85,81,102)(70,104,105,94)(71,103,74,95)(73,107,86,97)(75,109,110,99)(77,111,106,100)(80,112,90,101)(87,116,89,113)(92,117,108,115);(54,55,60,65)(56,61,67,83)(57,70,72,105)(58,75,78,110)(59,81,79,64)(62,86,88,73)(63,90,93,80)(66,94,96,104)(68,99,76,109)(69,102,98,85)(71,92,74,108)(77,87,106,89)(82,107,114,97)(84,112,91,101)(95,117,103,115)(100,116,111,113)Polycyclic presentation:
2^8,53|a^2=h,b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,f^d=fh,i^d=i^52554 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 324841);