Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #13622 |
294 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 3 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)(54,57,61,60)(55,63,66,83)(56,68,70,84)(58,72,78,77)(59,73,82,81)(62,85,86,88)(64,89,92,111)(65,79,94,91)(67,96,98,100)(69,101,104,115)(71,80,106,103)(74,108,95,113)(75,109,105,114)(76,110,102,112)(87,97,116,93)(90,107,99,117);(54,56)(55,62)(57,68)(58,75)(59,80)(60,84)(61,70)(63,85)(64,90)(65,93)(66,86)(67,95)(69,102)(71,81)(72,109)(73,106)(74,98)(76,104)(77,114)(78,105)(79,87)(82,103)(83,88)(89,107)(91,116)(92,99)(94,97)(96,113)(100,108)(101,112)(110,115)(111,117);(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)(56,67,70,98)(57,63)(58,74)(59,79)(60,83)(61,66)(62,69,86,104)(64,76)(65,81)(68,96,84,100)(71,107,106,117)(72,108)(73,94)(75,93,105,97)(77,113)(78,95)(80,99,103,90)(82,91)(85,101,88,115)(87,114,116,109)(89,110)(92,102)(111,112)Polycyclic presentation:
2^8,53|a^2=e,c^2=h,d^2=fgh,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^52,d^b=dfh,e^b=eh,f^b=fg,g^b=gh1196 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90031);