#27381549

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #20360

Subgroups

	subgroup	G/H
center	C₄	C₂²×C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#568	C₂
socle	C₂×C₅₃	C₂²×#95
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#568	D₅₃
hypocenter	C₅₃	#20360
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂×#568

1996 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

1850 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 167025);