#27402608

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1814 ⋊ C₂
Sylow	C₅₃ ⋊ #22694

Subgroups

	subgroup	G/H
center	C₄	#16811257
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1814	C₂
socle	C₂×C₅₃	D₄×C₄⋊C₄
prime radical	C₂×#13133237	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1814	D₅₃
hypocenter	C₅₃	#22694
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1814

812 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)(54,58)(55,64)(56,68)(57,70)(59,76)(60,75)(61,77)(62,79)(63,80)(65,84)(66,85)(67,82)(69,74)(71,73)(72,81)(78,83);(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)(54,57,59,73,61,74,60,68)(55,63,65,72,66,82,78,79)(56,58,70,76,71,77,69,75)(62,64,80,84,81,85,67,83);(54,56)(55,62)(57,58)(59,70)(60,69)(61,71)(63,64)(65,80)(66,81)(67,78)(68,75)(72,84)(73,76)(74,77)(79,83)(82,85);(54,55,60,65,61,66,59,78)(56,67,69,62,71,80,70,81)(57,72,68,82,74,79,73,63)(58,64,75,84,77,85,76,83)

Polycyclic presentation:

2^8,53|a^2=g,c^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,e^d=eh,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, 188084);