C₂×#72327612 #72339432

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂×#72327612

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

857112 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)(11,12,13,14)(15,16)(17,18)(19,22)(20,21);(15,17)(16,20)(18,22)(19,21);(8,10)(12,14);(1,2);(4,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22);(15,18)(16,21)(17,22)(19,20);(3,4,5,6)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72327629
maximal normal subgroup of	C₂×#72327629

GAP

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

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

C2×#72327612 #72339432