C₂²×#14754 #72321562

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#14754

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

75541 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#4306369, C₂²×#773894, C₂²×#687082
maximal normal subgroup of	C₂²×#4306369, C₂²×#773894, C₂²×#687082
maximal quotient of	C₂²×#154635

GAP

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

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

C22×#14754 #72321562