C₂²×#14419 #72289239

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶×C₄

Aliases

direct product

C₂²×#14419

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

111525 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#846785, C₂²×#1418385, C₂²×#534306, …
maximal normal subgroup of	C₂²×#846785, C₂²×#1418385, C₂²×#534306, …
maximal quotient of	C₂³×#14419, C₂×#72298516

GAP

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

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

C22×#14419 #72289239