C₅₃×D₄₅ #18012008

order	4770
isoclinism	isoclinic class of D₄₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅₃×D₄₅
unambiguous semidirect product	C₅⋊(D₉×C₅₃)
unambiguous semidirect product	C₉⋊(D₅×C₅₃)

Splittings

Fitting, Hall	C₅×C₉×C₅₃ ⋊ C₂
derived, Hall	C₅×C₉ ⋊ C₂×C₅₃
Hall	C₉×C₅₃ ⋊ D₅
Hall	C₅×C₅₃ ⋊ D₉
Sylow	C₉ ⋊ D₅×C₅₃
Sylow	C₅ ⋊ D₉×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₄₅
commutator, cosocle, hypocenter	C₅×C₉	C₂×C₅₃
Frattini	C₃	D₁₅×C₅₃
Fitting	C₅×C₉×C₅₃	C₂
socle	C₃×C₅×C₅₃	D₃
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₅₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 2385

A minimal faithful permutation representation — 2 generators on 67 points:

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

Conjugacy classes & character table

1272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅₃×D₂₂₅, C₅×C₅₃×D₄₅, C₅₃×C₅⋊D₄₅, …
maximal normal subgroup of	C₅×C₅₃×D₄₅, C₇×C₅₃×D₄₅
maximal quotient of	C₅₃×D₂₂₅, C₅×C₅₃×D₄₅, C₅₃×C₅⋊D₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4770, 7);

C53×D45 #18012008