Finite Group Databasebeta

← Prev (#64616) Next (#64618) →

C₄×D₃×C₁₃ #64617

order	312
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×C₃×C₄×D₃ (order 288) · Inn(G) = D₃ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₂×C₄×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

C₄×D₃×C₁₃

Splittings

Fitting	C₃×C₄×C₁₃ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₄×C₁₃
Hall	C₃×C₁₃ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₁₃	D₃
commutator, hypocenter	C₃	C₂×C₄×C₁₃
Frattini	C₂	C₂×D₃×C₁₃
Fitting	C₃×C₄×C₁₃	C₂
socle	C₂×C₃×C₁₃	C₂²
prime radical	C₂×D₃×C₁₃	C₂
cosocle	C₂×C₃	C₂²×C₁₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₄×C₁₃

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1055, #10597593, D₃×C₈×C₁₃, …
maximal normal subgroup of	D₃×#1055, #10597593, D₃×C₈×C₁₃, …
maximal quotient of	D₃×C₈×C₁₃, C₁₃×ΓL(1,𝔽₂₅), C₄×Dic₃×C₁₃, …
automorphism group of	C₂²×C₅₃, D₃×C₅₃
outer automorphism group of	C₂²×C₅₃, Q₈×C₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 33);