Finite Group Databasebeta

← Prev (#72226597) Next (#72226599) →

C₂×C₄×#87830 #72226598

order	3200
isoclinism	isoclinic class of D₅⋊AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

C₂×C₄×#87830

Splittings

C₅² ⋊ C₂×C₄³

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₅⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅²	C₂×C₄³
Frattini	C₂²	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅²	C₂×C₄
socle	C₂³×C₅²	C₂²×C₄
prime radical	C₂³×C₅⋊D₅	C₂³
cosocle	C₂²×C₅⋊D₅	C₂⁴
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂²×C₄

3042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72221487, C₂²×C₄×#87830, C₂×C₄²×D₅⋊AGL(1,𝔽₅)
maximal normal subgroup of	C₂×#72221487, C₂²×C₄×#87830, C₂×C₄²×D₅⋊AGL(1,𝔽₅)
maximal quotient of	C₄×#13103905, C₂×#72220669, #72242675, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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