C₂×#72268524 #72341814

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

Aliases

direct product

C₂×#72268524

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

772314 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#120×#525
maximal normal subgroup of	C₂×#120×#525

GAP

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

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

C2×#72268524 #72341814