C₂×#6342379 #72334459

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#6342379

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2847	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

85025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72237098, C₂×#72237103, C₂×#72236588, …
maximal normal subgroup of	C₂×#72237098, C₂×#72237103, C₂×#72236588, …
maximal quotient of	C₂×#72334308, C₂×#72339929

GAP

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

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

C2×#6342379 #72334459