C₂×#7671889 #72335165

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

Aliases

direct product

C₂×#7671889

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

109165 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)(9,12)(10,11)(13,14)(16,19)(20,22);(4,6)(8,11)(12,13)(17,21)(20,22);(4,6)(7,9)(8,12)(10,14)(11,13)(15,16,18,19)(17,22,21,20);(3,4)(5,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,20)(18,19)(21,22);(1,2);(3,4)(5,6)(15,17)(16,20)(18,21)(19,22);(7,8)(9,13)(10,11)(12,14)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236218
maximal normal subgroup of	C₂×#72236218
maximal quotient of	C₂×#72335117, C₂×#72335118, C₂×#72335130

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

C2×#7671889 #72335165