D₅₃×#2123 #27536826

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2123

Splittings

Fitting	C₅₃×#2123 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2123

Subgroups

	subgroup	G/H
center	C₂	C₂×#99×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2123	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	C₂×#41×D₅₃	C₂²
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	#2123	D₅₃
hypocenter	C₅₃	C₂×#2123
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2123

1214 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)(54,57,59,70)(55,61,63,77)(56,65,67,68)(58,69,72,85)(60,73,74,75)(62,76,71,84)(64,78,79,82)(66,80,81,83);(54,56,59,67)(55,60,63,74)(57,68,70,65)(58,66,72,81)(61,75,77,73)(62,64,71,79)(69,83,85,80)(76,82,84,78);(54,55)(56,64)(57,61)(58,71,72,62)(59,63)(60,66,74,81)(65,78)(67,79)(68,82)(69,84,85,76)(70,77)(73,80,75,83);(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)

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, 322302);

D53×#2123 #27536826