C₂×#7940997 #72304496

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

Aliases

direct product

C₂×#7940997

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 10578 conjugacy classes of subgroups; only the 31893 normal subgroups are shown.

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72304329
maximal normal subgroup of	C₂×#72304329

GAP

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

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

C2×#7940997 #72304496