D₄×#2404 #72264104

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

D₄×#2404

Subgroups

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

65777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(2,4)(6,8)(14,18)(16,20);(5,6)(7,8)(10,12)(13,14,17,18)(15,16,19,20);(9,10)(11,12);(5,6)(7,8)(13,14,17,18)(15,16,19,20);(13,15)(14,16)(17,19)(18,20);(1,2)(3,4)(5,6)(7,8)(13,16)(14,19)(15,18)(17,20)

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#34736, D₄×#35346
maximal normal subgroup of	D₄×#34736, D₄×#35346
maximal quotient of	#72264851, #72336443, #72336455, …

GAP

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

gap> G := Group((2,4)(6,8)(14,18)(16,20), (5,6)(7,8)(10,12)(13,14,17,18)(15,16,19,20), (9,10)(11,12), (5,6)(7,8)(13,14,17,18)(15,16,19,20), (13,15)(14,16)(17,19)(18,20), (1,2)(3,4)(5,6)(7,8)(13,16)(14,19)(15,18)(17,20));

D4×#2404 #72264104