D₈×#365 #72309596

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₈×#365

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#106
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×D₈	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309657, D₈×AGL(1,ℤ/16), #72310110, …
maximal normal subgroup of	#72309657, D₈×AGL(1,ℤ/16), #72310110, …

GAP

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

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

D8×#365 #72309596