C₂²×#16291 #72296934

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂²×#16291

Subgroups

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

85125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#1572498, C₂²×#3825309, C₂²×#3858074, …
maximal normal subgroup of	C₂²×#1572498, C₂²×#3825309, C₂²×#3858074, …
maximal quotient of	C₂²×#158317

GAP

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

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

C22×#16291 #72296934