D₅₃×#2121 #27536815

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

Aliases

direct product

D₅₃×#2121

Splittings

Fitting	C₅₃×#2121 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2121

Subgroups

	subgroup	G/H
center	C₂	C₂×#99×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2121	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	#534×D₅₃	C₂
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	#2121	D₅₃
hypocenter	C₅₃	C₂×#2121
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ #2121

1494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 4 generators on 69 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)(56,63)(59,67)(62,68)(64,69);(54,56)(55,59)(57,64)(58,63)(60,62)(61,67)(65,68)(66,69);(54,55)(56,62)(57,65,66,60)(58,61)(59,64,67,69)(63,68);(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, 322291);

D53×#2121 #27536815