D₅₃×#1659 #27365209

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

D₅₃×#1659

Splittings

Fitting	C₅₃×#1659 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1659

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1659	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	H(3,𝔽₄)×D₅₃	C₂
hypercenter	#1659	D₅₃
hypocenter	C₅₃	C₂×#1659
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1659

1150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 77 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)(54,55)(56,61)(57,59)(58,60)(65,75)(68,73)(70,77)(71,74);(56,60)(58,61)(62,64)(63,67)(65,74)(66,72)(68,77)(69,76)(70,73)(71,75);(55,58)(56,60)(59,61)(62,63,66,69)(64,70,72,77)(65,73,75,68)(67,74,76,71);(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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150685);

D53×#1659 #27365209