#27432226

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#455 ⋊ C₂
Sylow	C₅₃ ⋊ #25199

Subgroups

	subgroup	G/H
center	C₂	#20278247
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#455	C₂
socle	C₂×C₅₃	#2424
prime radical	#20284072	C₂
hypercenter	C₂×#455	D₅₃
hypocenter	C₅₃	#25199
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#455

990 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)(54,58,61,57)(55,63,65,81)(56,67,69,71)(59,72,60,74)(62,78,80,77)(64,70,75,73)(66,82,79,85)(68,84,76,83);(54,57,61,58)(55,63,65,81)(56,67,69,71)(59,72,60,74)(62,77,80,78)(64,73,75,70)(66,82,79,85)(68,83,76,84);(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)(54,56,60,68,61,69,59,76)(55,62,64,79,65,80,75,66)(57,71,72,83,58,67,74,84)(63,78,70,85,81,77,73,82);(54,55,60,64,61,65,59,75)(56,66,68,62,69,79,76,80)(57,70,72,81,58,73,74,63)(67,78,84,85,71,77,83,82)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 217702);