C₂×#674660 #72335207

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂×#674660

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

75805 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)(12,13)(15,16,18,20)(17,21,22,19);(8,9)(12,13)(17,22)(19,21);(4,6)(11,12)(13,14)(15,16)(17,21)(18,20)(19,22);(4,6)(12,13)(15,16)(17,21)(18,20)(19,22);(1,2);(3,4)(5,6)(7,9,10,8)(11,13,14,12)(15,17)(16,19)(18,22)(20,21)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72335133, C₂×#72335135
maximal normal subgroup of	C₂×#72335133, C₂×#72335135
maximal quotient of	C₂×#72335143

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

C2×#674660 #72335207