C₂²×#34175 #72324187

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂²×#34175

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

316869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7631202, C₂²×#7633038, C₂²×C₂²⋊C₄×H(5,𝔽₂), …
maximal normal subgroup of	C₂²×#7631202, C₂²×#7633038, C₂²×C₂²⋊C₄×H(5,𝔽₂), …
maximal quotient of	C₂²×#531611, C₂²×D₄×#380, C₂²×#520481, …

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

C22×#34175 #72324187