C₂×#142539 #72260696

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#142539

Subgroups

	subgroup	G/H
center, socle	C₂³	#2021
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	C₂×#3490	C₂²
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

4659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

(3,4,6,5,7,9,10,8)(11,12)(13,19,21,25)(14,22,23,18)(15,17)(16,20)(24,26);(3,5,6,4,7,8,10,9)(11,13,14,20,15,21,23,26)(12,16,22,19,17,24,18,25);(4,8,9,5)(12,17)(20,26);(1,2)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233789
maximal normal subgroup of	C₂×#72233789
maximal quotient of	#72261963

GAP

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

gap> G := Group((3,4,6,5,7,9,10,8)(11,12)(13,19,21,25)(14,22,23,18)(15,17)(16,20)(24,26), (3,5,6,4,7,8,10,9)(11,13,14,20,15,21,23,26)(12,16,22,19,17,24,18,25), (4,8,9,5)(12,17)(20,26), (1,2));

C2×#142539 #72260696