#27290201

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10900

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133186
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#2873	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#2873	D₅₃
hypocenter	C₅₃	#10900
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2873

686 conjugacy classes of subgroups.

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)(54,56,57,55)(58,61)(59,66)(60,70)(62,75)(63,76)(64,77)(65,79)(67,82)(68,83)(69,84)(71,86)(72,88)(73,87)(74,78)(80,89)(81,85);(58,60,63,71,64,73,62,72)(59,65,67,80,68,81,78,69)(61,70,76,86,77,87,75,88)(66,79,82,89,83,85,74,84);(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)(54,55,57,56)(58,59)(60,69)(61,74)(62,78)(63,67)(64,68)(65,71)(66,76)(70,85)(72,81)(73,80)(75,83)(77,82)(79,87)(84,86)(88,89)

Polycyclic presentation:

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

Conjugacy classes & character table

1832 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 75677);