D₅₃×#1702 #27366752

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

D₅₃×#1702

Splittings

Fitting	C₅₃×#1702 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1702

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1702	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#127×D₅₃	C₂²
hypercenter	#1702	D₅₃
hypocenter	C₅₃	C₂×#1702
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1702

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

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)(55,60)(58,61)(62,64,65,71)(63,68,67,76)(66,75,73,72)(69,77,74,70);(54,55,59,61,57,60,56,58)(63,67)(68,76)(69,74)(70,77);(55,58)(56,59)(60,61)(62,63)(64,70)(65,67)(66,74)(68,72)(69,73)(71,77)(75,76);(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, 152228);

D53×#1702 #27366752