D₅₃×AGL(1,ℤ/16) #27537529

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

Aliases

direct product

D₅₃×AGL(1,ℤ/16)

Splittings

Fitting	C₅₃×AGL(1,ℤ/16) ⋊ C₂
Sylow	C₅₃ ⋊ C₂×AGL(1,ℤ/16)

Subgroups

	subgroup	G/H
center	C₂	#442×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×AGL(1,ℤ/16)	C₂
socle	C₂×C₅₃	C₂×#442
prime radical	#509×D₅₃	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	AGL(1,ℤ/16)	D₅₃
hypocenter	C₅₃	C₂×AGL(1,ℤ/16)
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ AGL(1,ℤ/16)

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 69 points:

(1,3,11,14,5,12,4,10)(2,7,9,6,8,16,13,15)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19);(1,2)(3,7)(4,13)(5,8)(6,14)(9,11)(10,15)(12,16);(2,6,8,15)(3,10)(4,11)(7,13,16,9)(12,14);(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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)

Conjugacy classes & character table

644 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323005);

D53×AGL(1,ℤ/16) #27537529