C₃×C₂²⋊D₅₃ #11929670

order	1272
isoclinism	isoclinic class of Dic₁₀₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×C₂²⋊D₅₃

Splittings

Fitting	C₂²×C₃×C₅₃ ⋊ C₂
Hall	C₃×C₅₃ ⋊ D₄
Sylow	C₅₃ ⋊ C₃×D₄

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×D₅₃
commutator, cosocle	C₂×C₅₃	C₂²×C₃
Frattini	C₂	C₂×C₃×D₅₃
Fitting	C₂²×C₃×C₅₃	C₂
socle	C₂×C₃×C₅₃	C₂²
hypercenter	C₂²×C₃	D₅₃
hypocenter	C₅₃	C₃×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 636

A minimal faithful permutation representation — 2 generators on 60 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1272, 19);

C3×C22⋊D53 #11929670