C₂×#72323284 #72331176

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72323284

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 8032 conjugacy classes of subgroups; only the 45736 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)(12,14)(15,16,19,18)(17,20,22,21);(7,8)(9,10)(11,12)(13,14)(17,22)(20,21);(15,17)(16,20)(18,21)(19,22);(1,2);(4,6)(8,10);(15,17)(16,21)(18,20)(19,22);(3,4)(5,6)(7,8)(9,10)(15,18,19,16)(17,21,22,20)

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

C2×#72323284 #72331176