C₂×#612462 #72341734

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

Aliases

direct product

C₂×#612462

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

60547 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72268519, C₂×#72303259, C₂×#72268523
maximal normal subgroup of	C₂×#72268519, C₂×#72303259, C₂×#72268523

GAP

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

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

C2×#612462 #72341734