D₄×#12068 #72319583

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₄×#12068

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
socle	C₂²	C₂²×#1580
prime radical	D₄×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×D₄ ◁ D₄×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×AGL(1,ℤ/32), S₄×#12068
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((5,6,8,15,10,17,27,13,12,18,25,16,11,29,9,28)(7,19,21,14,20,31,22,32,24,34,26,33,23,30,35,36), (1,2)(3,4), (2,4), (6,13,17,16,18,28,29,15)(7,20,24,23)(8,25)(9,27)(14,30,36,34,33,31,32,19)(21,35,26,22), (5,7,8,26,10,20,27,35,12,24,25,21,11,23,9,22)(6,14,13,19,29,36,15,30,18,33,28,34,17,32,16,31));

D4×#12068 #72319583