C₂×#7677051 #72335167

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

Aliases

direct product

C₂×#7677051

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

109501 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:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236218
maximal normal subgroup of	C₂×#72236218
maximal quotient of	C₂×#72297694, C₂×#72297197, C₂×#72297520

GAP

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

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

C2×#7677051 #72335167