C₂×#13104006 #72226397

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13104006

Splittings

Sylow

C₅² ⋊ C₂×#389

Subgroups

	subgroup	G/H
center	C₂³	C₂×D₅⋊AGL(1,𝔽₅)
commutator	C₂²×C₅²	C₂³×C₄
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
hypercenter	C₂²×C₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₂×#389
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

2018 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:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72226398

GAP

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

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

C2×#13104006 #72226397