D₅₃×#2223 #27538657

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

Aliases

direct product

D₅₃×#2223

Splittings

Fitting	C₅₃×#2223 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2223

Subgroups

	subgroup	G/H
center	C₂	#466×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2223	C₂
socle	C₂×C₅₃	C₂×#466
prime radical	#472×D₅₃	C₂
hypercenter	#2223	D₅₃
hypocenter	C₅₃	C₂×#2223
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2223

658 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:

(1,3)(2,10)(4,15)(5,19)(6,14)(7,16)(8,20)(9,24)(11,28)(12,26)(13,27)(17,29)(18,30)(21,32)(22,25)(23,31)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,2,9,21,6,12,4,8,7,13,17,22,5,11,18,23)(3,10,24,32,14,26,15,20,16,27,29,25,19,28,30,31);(2,8)(3,14)(4,17)(5,6)(9,18)(10,25)(11,22)(12,21)(13,23)(15,30)(16,19)(20,26)(27,32)(28,31);(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)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324133);

D53×#2223 #27538657