D₄×#2403 #72232046

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

D₄×#2403

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

112833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 16 points:

(5,6,8,7)(9,10,12,11)(13,14)(15,16);(5,6,8,7)(9,10,12,11);(5,6,8,7)(9,10,12,11)(13,14,15,16);(2,4)(9,10)(11,12);(6,7)(10,11)(13,14)(15,16);(1,2)(3,4)(5,7,8,6)(9,11,12,10)

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232064, #72233580, D₄²×#120, …
maximal normal subgroup of	#72232064, #72233580, D₄²×#120, …
maximal quotient of	D₄×#23882, C₂×D₄×#2403, #72298594, …

GAP

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

gap> G := Group((5,6,8,7)(9,10,12,11)(13,14)(15,16), (5,6,8,7)(9,10,12,11), (5,6,8,7)(9,10,12,11)(13,14,15,16), (2,4)(9,10)(11,12), (6,7)(10,11)(13,14)(15,16), (1,2)(3,4)(5,7,8,6)(9,11,12,10));

D4×#2403 #72232046