#27431413

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2046 ⋊ C₂
Sylow	C₅₃ ⋊ #25145

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2046	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20283959	C₂
hypercenter	#2046	D₅₃
hypocenter	C₅₃	#25145
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2046

996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 89 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 216889);