D₅₃×#2031 #27411512

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅₃×#2031

Splittings

Fitting	C₅₃×#2031 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2031

Subgroups

	subgroup	G/H
center	C₄	#124×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2031	C₂
socle	C₂×C₅₃	C₂×#393
prime radical	C₂×#41×D₅₃	C₂²
hypercenter	#2031	D₅₃
hypocenter	C₅₃	C₂×#2031
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2031

1234 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:

(1,3)(2,7)(4,11)(5,13)(6,12)(8,16)(9,15)(10,14)(17,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19);(3,11,12,13)(7,10,15,16);(1,2)(3,10,12,16)(4,8)(5,14)(6,9)(7,13,15,11);(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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196988);

D53×#2031 #27411512