Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #13638 |
182 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 182 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)(54,57,60,75)(55,62,66,65)(56,69,71,101)(58,73,79,109)(59,74,81,77)(61,83,84,86)(63,87,90,89)(64,88,91,80)(67,92,94,96)(68,98,99,78)(70,100,104,113)(72,102,106,103)(76,108,110,82)(85,93,112,111)(95,114,116,107)(97,115,117,105);(54,56,68,97,59,72,58,70,75,101,78,105,77,103,109,113,60,71,99,117,81,106,79,104,57,69,98,115,74,102,73,100)(55,61,82,107,64,85,63,67,62,83,76,95,88,93,87,92,66,84,108,114,91,112,90,94,65,86,110,116,80,111,89,96);(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)(56,67,71,94)(57,62)(58,76)(59,80)(60,66)(61,70,84,104)(63,78)(64,74)(65,75)(68,87)(69,92,101,96)(72,107,106,114)(73,108)(77,91)(79,110)(81,88)(82,109)(83,100,86,113)(85,97,112,117)(89,99)(90,98)(93,115,111,105)(95,103,116,102)Polycyclic presentation:
2^8,53|a^2=e,b^2=dfg,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,e^b=eh2132 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90089);