C₂×#72323404 #72333196

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

Aliases

direct product

C₂×#72323404

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 15572 conjugacy classes of subgroups; only the 38248 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)(12,14);(11,12)(13,14)(15,16)(17,21)(18,19)(20,22);(15,17)(16,20)(18,21)(19,22);(1,2);(4,6)(8,10);(7,8)(9,10)(15,17)(16,21)(18,20)(19,22);(3,4)(5,6)(15,18,19,16)(17,21,22,20)

Conjugacy classes & character table

308 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)(12,14), (11,12)(13,14)(15,16)(17,21)(18,19)(20,22), (15,17)(16,20)(18,21)(19,22), (1,2), (4,6)(8,10), (7,8)(9,10)(15,17)(16,21)(18,20)(19,22), (3,4)(5,6)(15,18,19,16)(17,21,22,20));

C2×#72323404 #72333196