C₄×#12366 #72258289

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×#12366

Subgroups

	subgroup	G/H
center	C₂×C₄	#1629
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂²	C₂×#1629
prime radical	C₂×#3292	C₂²
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

1897 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:

(5,6,9,13)(7,10,17,20)(8,11,18,14)(12,15,19,16);(6,10,13,20)(7,15)(8,18)(11,12,14,19)(16,17);(7,16)(8,18)(10,12)(11,14)(15,17)(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)×#12366
maximal normal subgroup of	C₄×#489347
maximal quotient of	AGL(1,𝔽₅)×#12366

GAP

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

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

C4×#12366 #72258289