D₅₃×#2216 #27538512

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

Aliases

direct product

D₅₃×#2216

Splittings

Fitting	C₅₃×#2216 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2216

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2216	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×D₅₃	D₈
hypercenter	#2216	D₅₃
hypocenter	C₅₃	C₂×#2216
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 points:

(2,4)(37,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39);(5,7,16,32,9,18,22,24,10,19,27,36,21,33,8,17)(6,11,20,30,13,25,28,31,14,26,35,34,23,29,12,15);(1,2)(3,4)(5,6,10,14)(7,15,19,31)(8,20,22,35)(9,23,21,13)(11,24,26,17)(12,27,28,16)(18,34,33,30)(25,32,29,36);(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)(86,87)(88,89)

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323988);

D53×#2216 #27538512