D₄×#2849 #72334289

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

D₄×#2849

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	C₂×D₄×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

72669 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:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#34980, C₂×D₄×#2849
maximal normal subgroup of	D₄×#34980, C₂×D₄×#2849
maximal quotient of	D₄×#14694, C₂×D₄×#2849, #72334977, …

GAP

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

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

D4×#2849 #72334289