Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #33827 |
464 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 125 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)(54,56,57,60)(55,59,58,61)(64,75)(69,90)(74,101)(76,104)(77,105)(89,116)(91,118)(92,109)(100,123)(102,115)(103,110)(106,122)(107,125)(108,119)(117,121)(120,124);(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)(55,58)(59,61)(62,65,68,81)(63,70,73,95)(64,74,75,101)(66,83,86,114)(67,88,84,80)(69,89,90,116)(71,97,98,113)(72,99,87,94)(76,107,104,125)(77,110,105,103)(78,111,112,82)(79,96,93,85)(91,120,118,124)(92,121,109,117)(100,122,123,106)(102,119,115,108);(54,55)(56,59)(57,58)(60,61)(62,64,68,75)(63,69,73,90)(65,74,81,101)(66,76,86,104)(67,77,84,105)(70,89,95,116)(71,91,98,118)(72,92,87,109)(78,100,112,123)(79,102,93,115)(80,103,88,110)(82,106,111,122)(83,107,114,125)(85,108,96,119)(94,117,99,121)(97,120,113,124);(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)(55,58)(56,60)(62,63,68,73)(64,69,75,90)(65,78,81,112)(66,82,86,111)(67,87,84,72)(70,93,95,79)(71,96,98,85)(74,100,101,123)(76,106,104,122)(77,109,105,92)(80,113,88,97)(83,99,114,94)(89,115,116,102)(91,119,118,108)(103,124,110,120)(107,121,125,117)Polycyclic presentation:
2^8,53|a^2=h,b^2=h,c^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,i^a=i^52,c^b=ce,d^b=dh,f^c=fg,g^c=gh,i^c=i^521496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 326000);