C₂×#72334296 #72334556

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72334296

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 11514 conjugacy classes of subgroups; only the 43088 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

332 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₄×#34980
maximal normal subgroup of	C₂×D₄×#34980

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

C2×#72334296 #72334556