C₂×#7631296 #72304244

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

Aliases

direct product

C₂×#7631296

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

92073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

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

C2×#7631296 #72304244