C₂²×#13674 #72306444

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#13674

Subgroups

	subgroup	G/H
center, socle	C₂³	#2196
commutator, Frattini, cosocle	#127	C₂⁵
LCS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#127 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

9385 conjugacy classes of subgroups; only the 589 normal subgroups are shown.

normal subgroups only

This group has 9385 conjugacy classes of subgroups — too many to draw. Only its 589 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#512862
maximal normal subgroup of	C₂²×#512862
maximal quotient of	#72338401, #72338404

GAP

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

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