Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #11869 |
364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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)(54,57,59,76,61,78,60,77)(55,63,85,108,66,91,65,73)(56,68,71,74,70,101,72,102)(58,81,83,112,82,75,84,111)(62,86,95,116,88,114,87,89)(64,93,96,107,94,90,79,113)(67,97,106,115,98,109,92,99)(69,103,104,117,105,100,80,110);(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)(54,56,61,70)(55,62,66,88)(57,74,78,102)(58,80,82,104)(59,72,60,71)(63,89,91,116)(64,92,94,106)(65,95,85,87)(67,96,98,79)(68,77,101,76)(69,83,105,84)(73,86,108,114)(75,100,81,103)(90,97,93,109)(99,107,115,113)(110,111,117,112);(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)(57,73)(58,79,82,96)(59,65)(60,85)(61,66)(62,69)(63,77)(64,83,94,84)(68,99,101,115)(71,92,72,106)(74,109,102,97)(75,93,81,90)(76,91)(78,108)(80,95)(86,110)(87,104)(88,105)(89,103)(100,116)(107,111,113,112)(114,117)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,c^2=gh,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,i^a=i^52,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,i^b=i^52,d^c=dh,e^d=eh1178 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 80567);