D₅₃×#2048 #27411918

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2048

Splittings

Fitting	C₅₃×#2048 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2048

Subgroups

	subgroup	G/H
center	C₂	#394×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2048	C₂
socle	C₂×C₅₃	C₂×#394
prime radical	C₂×AGL(1,ℤ/8)×D₅₃	C₂
hypercenter	#2048	D₅₃
hypocenter	C₅₃	C₂×#2048
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2048

1450 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:

(1,3,4,11,6,12,5,10)(2,8,7,14,9,15,13,16)(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)(3,8)(4,13)(5,7)(6,9)(10,14)(11,16)(12,15);(2,7)(3,10)(4,5)(8,15)(9,13)(11,12);(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, 197394);

D53×#2048 #27411918