C₂×#903442 #72297289

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#903442

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

46931 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232261, C₂×#72298099, C₂×#72298081
maximal normal subgroup of	C₂×#72232261, C₂×#72298099, C₂×#72298081
maximal quotient of	C₂×#72278057

GAP

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

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

C2×#903442 #72297289