C₂×#120×#123 #72341873

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

Aliases

direct product

C₂×#120×#123

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂²×D₄×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

782502 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:

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

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

C2×#120×#123 #72341873