C₂²×#6454041 #72308692

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂²×#6454041

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁷
socle	C₂⁴	C₂²×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

599708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72308187

GAP

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

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

C22×#6454041 #72308692