#27530968

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1820 ⋊ C₂
Sylow	C₅₃ ⋊ #32701

Subgroups

	subgroup	G/H
center	C₂	#20282092
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1820	C₂
socle	C₂×C₅₃	#2760
prime radical	#16811185	C₂²
hypercenter	#1820	D₅₃
hypocenter	C₅₃	#32701
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #1820

628 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 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)(54,58,59,72)(55,62,63,77)(56,66,67,80)(57,70,71,83)(60,74,64,76)(61,75,68,81)(65,79,69,82)(73,84,78,85);(54,57)(55,61)(56,65,67,69)(58,70)(59,71)(60,73,64,78)(62,75)(63,68)(66,79,80,82)(72,83)(74,84,76,85)(77,81);(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)(55,60)(57,69)(58,66)(59,67)(61,73)(62,76)(63,64)(65,71)(68,78)(70,82)(72,80)(74,77)(75,85)(79,83)(81,84);(54,55)(56,64,67,60)(57,68,71,61)(58,62)(59,63)(65,78)(66,74,80,76)(69,73)(70,81,83,75)(72,77)(79,84)(82,85)

Polycyclic presentation:

2^8,53|a^2=fg,c^2=ef,d^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=de,f^b=fh,i^b=i^52,g^c=gh

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 316444);