C₄×#12765 #72282217

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₄×#12765

Subgroups

	subgroup	G/H
center	C₄²	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄²
Frattini, cosocle	C₂²×#41	C₂⁴
socle	C₂²	C₂×#1790
prime radical	C₂×#3481	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

5934 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#511819, #72312952
maximal normal subgroup of	C₄×#511819, #72312952

GAP

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

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

C4×#12765 #72282217