C₂×#72232294 #72330951

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂×#72232294

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁷
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 7722 conjugacy classes of subgroups; only the 445304 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72323292

GAP

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

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

C2×#72232294 #72330951