Finite Group Databasebeta

← Prev (#6813) Next (#6815) →

C₂×Dic₃₀ #6814

order	240
isoclinism	isoclinic class of Dic₃₀
automorphism	Aut(G) = D₃×AGL(1,𝔽₅)×B(4,𝔽₂) (order 7680) · Inn(G) = C₂×D₁₅ · Out(G) = C₄×#120
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

C₂×Dic₃₀

Splittings

Hall	C₃×C₅ ⋊ C₂×Q₈
Sylow	C₅ ⋊ C₂×Dic₆
Sylow	C₃ ⋊ C₂×Dic₁₀

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₁₅
commutator, cosocle	C₂×C₃×C₅	C₂³
Frattini	C₂	C₂²×D₁₅
Fitting	C₂×C₃×C₄×C₅	C₂
socle, prime radical	C₂²×C₃×C₅	C₂²
hypercenter	C₂×C₄	D₁₅
hypocenter	C₃×C₅	C₂×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#90963, #90974, #90975, …
maximal normal subgroup of	#90963, #90974, #90975, …
maximal quotient of	C₄×Dic₃₀, #91758, #91759, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 175);