#27302880

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#472 ⋊ C₂
Sylow	C₅₃ ⋊ #13319

Subgroups

	subgroup	G/H
center	C₂²	#16810756
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#472	C₂
socle	C₂²×C₅₃	#447
hypercenter	C₂×#472	D₅₃
hypocenter	C₅₃	#13319
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#472

572 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 89 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)(55,57)(58,60)(59,65)(61,75)(62,72)(63,73)(64,79)(66,85)(67,83)(68,84)(69,78)(70,77)(71,76)(74,87)(80,82)(81,89)(86,88);(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)(56,57)(59,64)(61,74)(62,76)(65,79)(66,67)(68,81)(69,82)(70,86)(71,72)(75,87)(77,88)(78,80)(83,85)(84,89);(58,59)(60,69)(61,64)(62,78)(63,68)(65,71)(66,74)(67,76)(70,82)(72,84)(73,83)(75,85)(77,81)(79,86)(80,88)(87,89)

Polycyclic presentation:

2^8,53|d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,g^b=gh,i^b=i^52,d^c=dh,e^c=eh

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88356);