C₂×#72323420 #72332933

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

Aliases

direct product

C₂×#72323420

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 14558 conjugacy classes of subgroups; only the 41088 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 18 points:

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

Conjugacy classes & character table

320 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((11,12)(13,14)(16,18), (15,16)(17,18), (7,8,10,9)(11,12,13,14), (1,2), (4,6)(11,12)(13,14), (8,9)(11,12)(13,14), (3,4)(5,6)(7,9,10,8));

C2×#72323420 #72332933