C₂×#72268520 #72341815

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

Aliases

direct product

C₂×#72268520

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 8900 conjugacy classes of subgroups; only the 43992 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:

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

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

C2×#72268520 #72341815