C₂²×#13877 #72268552

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

Aliases

direct product

C₂²×#13877

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×C₂²⋊C₄×#120, C₂²×#647670, C₂²×#612459, …
maximal normal subgroup of	C₂²×C₂²⋊C₄×#120, C₂²×#647670, C₂²×#612459, …
maximal quotient of	C₂×#72286388, C₂×#72286448, C₂³×#13877, …

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

C22×#13877 #72268552