C₂×#2805645 #72294669

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#2805645

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

60957 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293942, C₂×#72278043, C₂×#72277994, …
maximal normal subgroup of	C₂×#72293942, C₂×#72278043, C₂×#72277994, …
maximal quotient of	C₂×#72279098, C₂×#72278173

GAP

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

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

C2×#2805645 #72294669