C₂×#136537 #72258647

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

Aliases

direct product

C₂×#136537

Subgroups

	subgroup	G/H
center, socle	C₂³	#1629
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2271 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 22 points:

(3,4,6,5)(7,8)(9,12)(10,15)(11,16)(13,17)(14,19)(18,21)(20,22);(9,17)(10,19)(12,13)(14,15)(18,20)(21,22);(3,5,6,4)(8,12,16,22)(9,18)(10,19)(13,15,21,14)(17,20);(1,2)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213969
maximal normal subgroup of	C₂×#72213969

GAP

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

gap> G := Group((3,4,6,5)(7,8)(9,12)(10,15)(11,16)(13,17)(14,19)(18,21)(20,22), (9,17)(10,19)(12,13)(14,15)(18,20)(21,22), (3,5,6,4)(8,12,16,22)(9,18)(10,19)(13,15,21,14)(17,20), (1,2));

C2×#136537 #72258647