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C₉×C₁₉×AGL(1,𝔽₅) #16811827

order	3420
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₄×C₉×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

C₉×C₁₉×AGL(1,𝔽₅)

Splittings

Fitting, Hall	C₅×C₉×C₁₉ ⋊ C₄
derived, Sylow	C₅ ⋊ C₄×C₉×C₁₉
Hall	C₅×C₁₉ ⋊ C₄×C₉
Hall	C₅×C₉ ⋊ C₄×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₉×C₁₉	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₉×C₁₉
Frattini	C₃	C₃×C₁₉×AGL(1,𝔽₅)
Fitting	C₅×C₉×C₁₉	C₄
socle	C₃×C₅×C₁₉	C₃×C₄
prime radical	C₃×D₅×C₁₉	C₂×C₃
cosocle	C₃×D₅	C₂×C₃×C₁₉
LCS	G ▷ C₅
UCS	C₁ ◁ C₉×C₁₉

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 855

A minimal faithful permutation representation — 2 generators on 33 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)(30,31,32,33);(29,30,31,33,32)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₉×C₁₉×#1035, C₅×C₉×C₁₉×AGL(1,𝔽₅), C₉×C₁₉×#1043, …
maximal normal subgroup of	C₅×C₉×C₁₉×AGL(1,𝔽₅), C₇×C₉×C₁₉×AGL(1,𝔽₅), C₉×C₁₁×C₁₉×AGL(1,𝔽₅)
maximal quotient of	C₉×C₁₉×#1035, C₅×C₉×C₁₉×AGL(1,𝔽₅), C₉×C₁₉×#1043, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3420, 20);