C₂×#647923 #72323968

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

Aliases

direct product

C₂×#647923

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

55085 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)(11,12,13,14)(15,16,18,19)(17,21,22,20);(8,10)(16,19)(20,21);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(16,19)(17,22);(4,5)(15,17)(16,20)(18,22)(19,21);(1,2);(3,5,6,4)(15,16)(17,21)(18,19)(20,22)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232132, C₂×#72323252, C₂×#72281404, …
maximal normal subgroup of	C₂×#72232132, C₂×#72323252, C₂×#72281404, …
maximal quotient of	#72331189, #72331628, C₂²×#647923

GAP

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

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

C2×#647923 #72323968