#27469922

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2081 ⋊ C₂
Sylow	C₅₃ ⋊ #27989

Subgroups

	subgroup	G/H
center	C₂	#20279202
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2081	C₂
socle	C₂×C₅₃	#2504
prime radical	#20284362	C₂
hypercenter	#2081	D₅₃
hypocenter	C₅₃	#27989
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2081

654 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)(54,58,60,76,61,77,59,75)(55,63,67,81,65,79,64,74)(56,68,62,78,70,84,69,83)(57,71,72,85,73,82,66,80);(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)(55,62,65,69)(56,67,70,64)(58,71)(59,66)(60,72)(61,73)(63,78,79,83)(68,81,84,74)(75,80)(76,85)(77,82);(54,56,59,69,61,70,60,62)(55,57,64,66,65,73,67,72)(58,68,75,83,77,84,76,78)(63,71,74,80,79,82,81,85);(54,55,60,64,61,65,59,67)(56,66,62,73,70,72,69,57)(58,74,76,79,77,81,75,63)(68,82,78,85,84,71,83,80)

Polycyclic presentation:

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

Conjugacy classes & character table

638 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 255398);