C₃×D₂₅₃ #13095009

order	1518
isoclinism	isoclinic class of D₂₅₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₂₅₃
unambiguous semidirect product	C₁₁⋊(C₃×D₂₃)
unambiguous semidirect product	C₂₃⋊(C₃×D₁₁)

Splittings

Fitting, socle, Hall	C₃×C₁₁×C₂₃ ⋊ C₂
derived, Hall	C₁₁×C₂₃ ⋊ C₂×C₃
Hall	C₃×C₂₃ ⋊ D₁₁
Hall	C₃×C₁₁ ⋊ D₂₃
Sylow	C₂₃ ⋊ C₃×D₁₁
Sylow	C₁₁ ⋊ C₃×D₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₂₅₃
commutator, cosocle, hypocenter	C₁₁×C₂₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₁×C₂₃	C₂
LCS	G ▷ C₁₁×C₂₃
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 759

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3)(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);(5,14)(6,13)(7,12)(8,11)(9,10)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×D₁₁×D₂₃, D₃×D₂₅₃, D₃₃⋊D₂₃, …
maximal normal subgroup of	C₃×D₁₁×D₂₃, D₃×D₂₅₃, D₃₃⋊D₂₃, …
maximal quotient of	C₃×Dic₂₅₃, C₂×C₃×D₂₅₃, C₉×D₂₅₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1518, 7);

C3×D253 #13095009