C₂×#773763 #72334637

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

Aliases

direct product

C₂×#773763

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

74507 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233709, C₂×#72233707, C₂×#72233703
maximal normal subgroup of	C₂×#72233709, C₂×#72233707, C₂×#72233703
maximal quotient of	C₂×#72334399

GAP

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

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

C2×#773763 #72334637