C₂×#7764889 #72337789

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

Aliases

direct product

C₂×#7764889

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 6062 conjugacy classes of subgroups; only the 31925 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232275
maximal normal subgroup of	C₂×#72232275
maximal quotient of	C₂×#72232311

GAP

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

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

C2×#7764889 #72337789