C₂×#7631443 #72304186

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

Aliases

direct product

C₂×#7631443

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

97357 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)(3,4)(5,6)(15,16,20,18)(17,21,22,19);(3,4)(5,6)(7,8)(9,12)(10,13)(11,14)(15,17)(16,21)(18,19)(20,22);(7,9)(8,11)(10,14)(12,13);(3,4)(5,6)(15,18,20,16)(17,19,22,21);(4,6)(7,9)(8,12)(10,14)(11,13)(15,18)(16,20)(17,21)(19,22);(3,4)(5,6)(15,19,20,21)(16,22,18,17);(7,8)(9,11)(10,13)(12,14)

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72257036
maximal normal subgroup of	C₂×#72257036

GAP

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

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

C2×#7631443 #72304186