C₄×#12314 #72258284

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₄×#12314

Subgroups

	subgroup	G/H
center	C₂×C₄	#1619
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂²	C₂×#1619
prime radical	C₂×#12314	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#489347, AGL(1,𝔽₅)×#12314
maximal normal subgroup of	C₄×#489347
maximal quotient of	AGL(1,𝔽₅)×#12314

GAP

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

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

C4×#12314 #72258284