C₂×#4033838 #72294899

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

Aliases

direct product

C₂×#4033838

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

55329 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)(12,13)(15,16,19,17)(18,21,22,20);(4,6)(11,12,14,13)(16,17)(20,21);(8,10)(11,13,14,12)(16,17)(18,22);(7,8)(9,10)(15,17,19,16)(18,20,22,21);(3,4)(5,6)(15,18)(16,20)(17,21)(19,22);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293916, C₂×#72232261, C₂×#72277929, …
maximal normal subgroup of	C₂×#72293916, C₂×#72232261, C₂×#72277929, …
maximal quotient of	C₂×#72293941, C₂×#72278121

GAP

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

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

C2×#4033838 #72294899