C₂×C₅²×#444 #72253227

order	3200
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅² · Schur multiplier (H₂) = C₂⁵×C₅

Aliases

direct product

C₂×C₅²×#444

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅²	C₂×D₄
commutator	C₄	C₂³×C₄×C₅²
Frattini, cosocle	C₂×C₄	C₂⁴×C₅²
socle	C₂²×C₅²	C₄×D₄
prime radical	C₂×C₅²×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅² ◁ C₂²×C₅²×C₈ ◁ G

1856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Conjugacy classes & character table

1100 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	C₂×C₅⋊D₅×#444, C₂×#444×#1043
maximal subgroup of	C₂×C₅⋊D₅×#444
maximal normal subgroup of	C₂×C₅⋊D₅×#444

GAP

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

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

C2×C52×#444 #72253227