C₂×#72323289 #72331204

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂×#72323289

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 8472 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:

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

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

Relationships

maximal subgroup of	C₂×#72323292
maximal normal subgroup of	C₂×#72323292

GAP

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

gap> G := Group((7,8)(9,10)(11,12)(13,14)(15,16,19,20)(17,21,22,18), (7,8)(9,10)(12,13)(16,20)(17,22), (15,17)(16,21)(18,20)(19,22), (1,2), (4,6)(7,8,9,10)(11,13,14,12)(16,20)(18,21), (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×#72323289 #72331204