Finite Group Databasebeta

← Prev (#13103165) Next (#13103167) →

C₂×C₅²×#108 #13103166

order	1600
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅² · Schur multiplier (H₂) = C₂²×C₅

Aliases

C₂×C₅²×#108

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅²	D₄
commutator	C₄	C₂²×C₄×C₅²
Frattini, cosocle	C₂×C₄	C₂³×C₅²
socle	C₂²×C₅²	C₄⋊C₄
prime radical	C₂³×C₅²	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅² ◁ C₂²×C₄×C₅² ◁ G

424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2)(19,20,21,22,23);(3,4,7,10,8,11,6,16)(5,12,14,17,15,18,13,9)(24,25,26,27,28);(3,5,6,13,8,15,7,14)(4,9,10,12,11,17,16,18)

Conjugacy classes & character table

700 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#108×C₅⋊D₅, C₂×#108×#1043, C₂×#13102757, …
maximal subgroup of	C₂×#108×C₅⋊D₅, C₂×C₅²×#444, C₂×#13102757, …
maximal normal subgroup of	C₂×#108×C₅⋊D₅, C₂×C₅²×#444, C₂×#13102757, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5518);