D₄×#7389 #72319523

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

Aliases

direct product

D₄×#7389

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1333
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	D₄×#508	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

2657 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:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×AGL(1,ℤ/32), S₄×#7389
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((6,11,13,23,14,24,29,16)(7,12,17,27)(8,19)(9,22)(15,31,30,25,20,26,34,36)(18,33,32,28)(21,35), (1,2)(3,4), (2,4), (5,6,12,26,35,30,7,16,22,29,33,36,21,15,32,11,10,14,27,31,19,34,17,23,9,13,28,25,8,20,18,24));

D4×#7389 #72319523