D₅₃×#2237 #27538866

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

Aliases

direct product

D₅₃×#2237

Splittings

Fitting	C₅₃×#2237 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2237

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2237	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×D₅₃	D₈
hypercenter	#2237	D₅₃
hypocenter	C₅₃	C₂×#2237
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2237

586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 points:

(1,3,4,7)(2,6,5,8)(10,16)(12,23)(13,24)(15,27)(17,18)(19,32)(20,34)(21,35)(25,33)(26,31)(28,29)(30,39)(36,40)(37,38)(41,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43);(2,5)(6,8)(9,11,14,22)(10,15,19,30)(12,20,25,36)(13,21,24,35)(16,27,32,39)(17,28,31,38)(18,29,26,37)(23,34,33,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,19)(11,15,22,30)(12,16,25,32)(13,26,24,18)(17,33,31,23)(20,27,36,39)(21,37,35,29)(28,40,38,34);(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)(90,91)(92,93)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324342);

D53×#2237 #27538866