C₂²×#13561 #72306449

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#13561

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂⁴×C₄
Frattini, cosocle	#127	C₂⁵
prime radical	C₂³×#453	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×#127 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

14821 conjugacy classes of subgroups.

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#512862, C₂²×#152191, C₂²×#152018, …
maximal normal subgroup of	C₂²×#512862, C₂²×#152191, C₂²×#152018, …
maximal quotient of	#72338453, #72338455

GAP

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

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

C22×#13561 #72306449