C₂×#72337230 #72339957

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂×#72337230

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

4630390 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

#72339997, C₂×#72337391

GAP

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

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

C2×#72337230 #72339957