C₂×#612800 #72335213

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#612800

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

71617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297367, C₂×#72238732
maximal normal subgroup of	C₂×#72297367, C₂×#72238732
maximal quotient of	C₂×#72298546

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

C2×#612800 #72335213