C₂×#7819709 #72331570

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

Aliases

direct product

C₂×#7819709

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

82377 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)(7,8,11,13)(9,12,14,10)(15,16)(17,21)(18,22)(19,20);(4,6)(7,9,11,14)(8,12,13,10)(16,20)(18,21);(8,13)(10,12)(15,17)(16,21)(18,20)(19,22);(4,6)(15,18)(16,17)(19,21)(20,22);(1,2);(7,8)(9,10)(11,13)(12,14)(15,18)(16,17)(19,21)(20,22);(3,4,5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,21)(18,20)(19,22)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232293
maximal normal subgroup of	C₂×#72232293
maximal quotient of	C₂×#72323399

GAP

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

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

C2×#7819709 #72331570