C₂⁶×SD₁₆ #72313891

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂⁶×SD₁₆

Subgroups

	subgroup	G/H
center, socle	C₂⁷	D₄
commutator, Frattini, cosocle	C₄	C₂⁸
prime radical	C₂⁶×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

1012373 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁶×AGL(1,ℤ/8), C₂⁷×SD₁₆
maximal normal subgroup of	C₂⁶×AGL(1,ℤ/8), C₂⁷×SD₁₆
maximal quotient of	C₂⁶×#102, C₂⁶×#106, C₂⁷×SD₁₆

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

C26×SD16 #72313891