C₂×#7631293 #72304227

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7631293

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

85853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234467
maximal normal subgroup of	C₂×#72234467

GAP

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

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

C2×#7631293 #72304227