C₅₃×D₁₂₃ #27189341

order	13038
isoclinism	isoclinic class of D₁₂₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅₃×D₁₂₃
unambiguous semidirect product	C₃⋊(C₅₃×D₄₁)
unambiguous semidirect product	C₄₁⋊(D₃×C₅₃)

Splittings

Fitting, socle, Hall	C₃×C₄₁×C₅₃ ⋊ C₂
derived, Hall	C₃×C₄₁ ⋊ C₂×C₅₃
Hall	C₄₁×C₅₃ ⋊ D₃
Hall	C₃×C₅₃ ⋊ D₄₁
Sylow	C₄₁ ⋊ D₃×C₅₃
Sylow	C₃ ⋊ C₅₃×D₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₁₂₃
commutator, cosocle, hypocenter	C₃×C₄₁	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₃×C₄₁×C₅₃	C₂
LCS	G ▷ C₃×C₄₁
UCS	C₁ ◁ C₅₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 6519

A minimal faithful permutation representation — 2 generators on 97 points:

(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)(43,44)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(42,43,44)

Conjugacy classes & character table

3339 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₃×D₁₂₃, D₃×C₅₃×D₄₁, D₁₂₃⋊D₅₃, …
maximal normal subgroup of	D₅₃×D₁₂₃, D₃×C₅₃×D₄₁, D₁₂₃⋊D₅₃, …
maximal quotient of	C₅₃×Dic₁₂₃, C₂×C₅₃×D₁₂₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13038, 7);

C53×D123 #27189341