D₄×#7390 #72319666

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₄×#7390

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1333
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	C₂×D₄×D₁₆	C₄
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

3038 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:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×AGL(1,ℤ/32), S₄×#7390
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((6,12)(7,18)(8,23)(9,19)(10,24)(13,29)(14,27)(15,26)(16,31)(17,28)(20,35)(21,34)(22,33)(25,30)(32,36), (1,2)(3,4), (2,4), (5,6,7,13,8,14,20,27,24,29,35,12,9,25,18,26,11,17,22,31,19,32,34,36,10,16,21,28,23,15,33,30));

D4×#7390 #72319666