#27386677

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1932 ⋊ C₂
Sylow	C₅₃ ⋊ #21080

Subgroups

	subgroup	G/H
center	C₄	#121×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1932	C₂
socle	C₂×C₅₃	C₂×#388
prime radical	C₂×#41×D₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1932	D₅₃
hypocenter	C₅₃	#21080
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1932

1000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 172153);