C₂×#7686112 #72335169

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

Aliases

direct product

C₂×#7686112

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

109451 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:

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236218
maximal normal subgroup of	C₂×#72236218
maximal quotient of	C₂×#72335121, C₂×#72335133

GAP

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

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

C2×#7686112 #72335169