C₂×#8070226 #72332262

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

Aliases

direct product

C₂×#8070226

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

At least 10452 conjugacy classes of subgroups; only the 31829 normal subgroups are shown.

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323395

GAP

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

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

C2×#8070226 #72332262