#27466273

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2098 ⋊ C₂
Sylow	C₅₃ ⋊ #27693

Subgroups

	subgroup	G/H
center	C₄	#16811186
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2098	C₂
socle	C₂×C₅₃	#2487
prime radical	#16811169	C₂²
hypercenter	#2098	D₅₃
hypocenter	C₅₃	#27693
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2098

634 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)(54,58,59,73,61,77,60,69)(55,64,65,76,67,80,66,84)(56,57,70,74,71,75,78,79)(62,81,68,63,82,72,85,83);(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)(55,63)(56,69)(58,70)(59,74)(60,79)(61,75)(62,80)(64,82)(65,72)(66,81)(67,83)(68,84)(71,73)(76,85)(77,78);(54,56,59,70,61,71,60,78)(55,62,65,68,67,82,66,85)(57,73,74,77,75,69,79,58)(63,80,72,84,83,64,81,76);(54,55,60,65,61,67,59,66)(56,68,78,82,71,85,70,62)(57,72,79,83,75,81,74,63)(58,76,69,80,77,84,73,64)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 251749);