C₂×#72240162 #72333247

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

Aliases

direct product

C₂×#72240162

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 6462 conjugacy classes of subgroups; only the 47880 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323423
maximal normal subgroup of	C₂×#72323423

GAP

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

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

C2×#72240162 #72333247