C₂×#7670313 #72309222

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

Aliases

direct product

C₂×#7670313

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

76285 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72308109
maximal normal subgroup of	C₂×#72308109

GAP

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

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

C2×#7670313 #72309222