Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7423 |
140 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 41 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 — 2 generators on 117 points:
(1,2)(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)(54,56,61,69)(55,62,66,90)(57,74,76,100)(58,81,59,84)(60,87,82,72)(63,91,92,115)(64,94,95,117)(65,96,86,89)(67,97,98,73)(68,77,102,78)(70,75,71,85)(79,113,110,108)(80,114,83,88)(93,104,116,103)(99,106,105,112)(101,111,107,109);(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)(56,67,69,98)(57,73,76,97)(58,80)(59,83)(60,86)(61,66)(62,68)(63,77,92,78)(64,75)(65,82)(70,103,71,104)(72,105,87,99)(74,94)(79,112,110,106)(81,115,84,91)(85,95)(88,113)(89,107)(90,102)(93,111,116,109)(96,101)(100,117)(108,114)Polycyclic presentation:
2^8,53|a^2=d,b^2=h,c^2=fgh,e^2=g,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,i^a=i^52,c^b=cf,d^b=defgh,e^b=egh,f^b=fgh,g^b=gh,i^b=i^52,d^c=dh1175 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57139);