Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C53 ⋊ #33393 |
1102 conjugacy classes of subgroups; only the 391 normal subgroups are shown.
This group has 1102 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)(57,72)(62,85)(68,95)(73,100)(74,101)(81,107)(86,110)(87,88)(92,113)(96,115)(97,105)(102,112)(103,116)(104,117)(108,109)(111,114);(54,57)(55,62)(56,68)(58,73)(59,74)(60,72)(61,81)(63,86)(64,87)(65,88)(66,85)(67,92)(69,96)(70,97)(71,95)(75,102)(76,103)(77,104)(78,100)(79,105)(80,101)(82,108)(83,109)(84,107)(89,111)(90,112)(91,110)(93,114)(94,113)(98,116)(99,115)(106,117);(54,56,60,71)(55,61,66,84)(57,68,72,95)(58,76,78,98)(59,79,80,70)(62,81,85,107)(63,89,91,93)(64,83,65,82)(67,75,94,90)(69,77,99,106)(73,103,100,116)(74,105,101,97)(86,111,110,114)(87,109,88,108)(92,102,113,112)(96,104,115,117);(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)(54,55,59,64,60,66,80,65)(56,67,70,89,71,94,79,93)(57,62,74,87,72,85,101,88)(58,75,77,63,78,90,106,91)(61,76,82,69,84,98,83,99)(68,92,97,111,95,113,105,114)(73,102,104,86,100,112,117,110)(81,103,108,96,107,116,109,115)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,e^2=g,f^2=h,g^2=h,b^a=be,e^a=eg,g^a=gh,i^a=i^52,e^b=egh,f^b=fh,g^b=gh,d^c=dh1202 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 322539);