D₁₆×AGL(1,ℤ/8) #72342598

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₁₆×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁵
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×AGL(1,ℤ/8) ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

121 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×#509
maximal normal subgroup of	AGL(1,ℤ/8)×#509

GAP

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

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

D16×AGL(1,ℤ/8) #72342598