C₂×#7746091 #72302318

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

Aliases

direct product

C₂×#7746091

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

81077 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)(7,8)(9,11)(10,12)(13,14)(17,22)(18,21);(4,6)(9,14)(11,13)(16,20)(18,21);(3,4)(5,6)(7,9)(8,11)(10,13)(12,14)(15,16,19,20)(17,21,22,18);(3,4)(5,6)(7,10,12,8)(9,13,14,11)(15,17,19,22)(16,21,20,18);(3,4)(5,6)(7,11,12,13)(8,9,10,14)(15,17)(16,21)(18,20)(19,22);(3,4)(5,6)(7,9)(8,13)(10,11)(12,14)(15,18,19,21)(16,17,20,22);(7,11)(8,9)(10,14)(12,13)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72261361
maximal normal subgroup of	C₂×#72261361
maximal quotient of	C₂×#72279893, C₂×#72280033, C₂×#72280180, …

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

C2×#7746091 #72302318