D₅₃×#2005 #27410988

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₅₃×#2005

Splittings

Fitting	C₅₃×#2005 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2005

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2005	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#41×D₅₃	C₂²
hypercenter	#2005	D₅₃
hypocenter	C₅₃	C₂×#2005
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2005

1998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 73 points:

(2,4)(5,8,10,7)(6,12,14,15)(9,16,19,17)(11,13,20,18)(21,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23);(5,7,10,8)(6,11,14,20)(9,17,19,16)(12,18,15,13);(1,2)(3,4)(5,6)(7,15)(8,12)(9,18)(10,14)(11,16)(13,19)(17,20);(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)(70,71)(72,73)

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, 196464);

D53×#2005 #27410988