D₄×#7457 #72319497

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₄×#7457

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1368
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	C₂²×D₄	#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₄×C₈×D₄ ◁ D₄×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(6,11,12,21,13,22,25,24)(7,15,16,29)(8,18)(9,20)(14,28,26,23,27,35,34,36)(17,32,31,30)(19,33);(1,2)(3,4);(2,4);(5,6,7,11,8,14,15,23,9,12,17,24,18,26,30,28,10,13,16,22,19,27,29,36,20,25,31,21,33,34,32,35)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×AGL(1,ℤ/32), S₄×#7457
maximal normal subgroup of	D₄×AGL(1,ℤ/32)

GAP

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

gap> G := Group((6,11,12,21,13,22,25,24)(7,15,16,29)(8,18)(9,20)(14,28,26,23,27,35,34,36)(17,32,31,30)(19,33), (1,2)(3,4), (2,4), (5,6,7,11,8,14,15,23,9,12,17,24,18,26,30,28,10,13,16,22,19,27,29,36,20,25,31,21,33,34,32,35));

D4×#7457 #72319497