C₂×#72278357 #72320484

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72278357

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 8628 conjugacy classes of subgroups; only the 14768 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#120×#392, C₂²×#72278357
maximal normal subgroup of	C₂×#120×#392, C₂²×#72278357
maximal quotient of	C₂²×#72278357

GAP

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

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

C2×#72278357 #72320484