C₂²×#14762 #72289286

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂²×#14762

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

108045 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#773780, C₂²×#772794, C₂²×#647842, …
maximal normal subgroup of	C₂²×#773780, C₂²×#772794, C₂²×#647842, …
maximal quotient of	C₂²×#156706, C₂²×#154635, C₂²×#154647, …

GAP

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

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

C22×#14762 #72289286