C₂²×#14127 #72289153

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂²×#14127

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

204053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#612426, C₂²×#759288, C₂×#72287877, …
maximal normal subgroup of	C₂²×#612426, C₂²×#759288, C₂×#72287877, …
maximal quotient of	C₂×#72287877

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

C22×#14127 #72289153