C₂×#7696045 #72331444

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

Aliases

direct product

C₂×#7696045

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 10124 conjugacy classes of subgroups; only the 32109 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:

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

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232293
maximal normal subgroup of	C₂×#72232293
maximal quotient of	C₂×#72323384

GAP

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

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

C2×#7696045 #72331444