Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C53×#3028 ⋊ C2 |
| Sylow | C53 ⋊ #32897 |
3512 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 3512 conjugacy classes of subgroups — too many to draw. Only its 247 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)(55,64)(56,70)(57,74)(59,80)(60,77)(61,78)(62,83)(63,85)(65,90)(66,87)(67,88)(68,93)(69,94)(71,100)(72,101)(73,103)(75,105)(76,106)(79,99)(81,104)(82,107)(84,109)(86,111)(89,98)(91,110)(92,112)(95,116)(96,114)(97,115)(102,117)(108,113);(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)(54,57,61,76)(55,63,67,86)(56,69,72,97)(58,74,78,106)(59,75,60,81)(62,82,84,108)(64,85,88,111)(65,73,66,91)(68,92,89,102)(70,94,101,115)(71,95,79,96)(77,104,80,105)(83,107,109,113)(87,110,90,103)(93,112,98,117)(99,114,100,116);(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)(54,56)(55,62)(57,69)(58,70)(59,79)(60,71)(61,72)(63,82)(64,83)(65,89)(66,68)(67,84)(73,102)(74,94)(75,96)(76,97)(77,100)(78,101)(80,99)(81,95)(85,107)(86,108)(87,93)(88,109)(90,98)(91,92)(103,117)(104,116)(105,114)(106,115)(110,112)(111,113);(54,55,61,67)(56,68,72,89)(57,73,76,91)(58,64,78,88)(59,66,60,65)(62,79,84,71)(63,81,86,75)(69,82,97,108)(70,98,101,93)(74,103,106,110)(77,90,80,87)(83,100,109,99)(85,104,111,105)(92,95,102,96)(94,113,115,107)(112,114,117,116)Polycyclic presentation:
2^8,53|a^2=h,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,g^b=gh,i^b=i^52,f^c=fh,i^c=i^52,e^d=eh800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 318332);