C₂×D₄×#2849 #72334544

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×D₄×#2849

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂²×D₄×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

983968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

(3,4,5,6)(7,8,9,10)(12,14)(17,21)(19,22);(4,6)(7,8,9,10)(11,12)(13,14)(15,16,18,20)(17,22,21,19);(12,14);(11,12)(13,14);(1,2);(3,4)(5,6)(7,8)(9,10);(15,17)(16,19)(18,21)(20,22)

Conjugacy classes & character table

440 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₄×#34980
maximal normal subgroup of	C₂×D₄×#34980

GAP

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

gap> G := Group((3,4,5,6)(7,8,9,10)(12,14)(17,21)(19,22), (4,6)(7,8,9,10)(11,12)(13,14)(15,16,18,20)(17,22,21,19), (12,14), (11,12)(13,14), (1,2), (3,4)(5,6)(7,8)(9,10), (15,17)(16,19)(18,21)(20,22));

C2×D4×#2849 #72334544