Finite Group Databasebeta

← Prev (#90709) Next (#90711) →

C₄×D₃×C₁₉ #90710

order	456
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂³×D₃×C₉ (order 432) · Inn(G) = D₃ · Out(G) = C₂³×C₉
homology	Abelianization (H₁) = C₂×C₄×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

C₄×D₃×C₁₉

Splittings

Fitting	C₃×C₄×C₁₉ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₄×C₁₉
Hall	C₃×C₁₉ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₁₉	D₃
commutator, hypocenter	C₃	C₂×C₄×C₁₉
Frattini	C₂	C₂×D₃×C₁₉
Fitting	C₃×C₄×C₁₉	C₂
socle	C₂×C₃×C₁₉	C₂²
prime radical	C₂×D₃×C₁₉	C₂
cosocle	C₂×C₃	C₂²×C₁₉
LCS	G ▷ C₃
UCS	C₁ ◁ C₄×C₁₉

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 228

A minimal faithful permutation representation — 2 generators on 26 points:

(2,3)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(1,2,3)(4,5,6,7)

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₁₉⋊C₈, #11744816, D₃×C₈×C₁₉, …
maximal normal subgroup of	D₃×C₁₉⋊C₈, #11744816, D₃×C₈×C₁₉, …
maximal quotient of	D₃×C₈×C₁₉, C₁₉×ΓL(1,𝔽₂₅), C₄×Dic₃×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 30);