C₂²×#63012 #72278967

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#63012

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

517117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#10586472, C₂²×#10586564, C₂³×#63012
maximal normal subgroup of	C₂²×#10586472, C₂²×#10586564, C₂³×#63012
maximal quotient of	C₂²×#7684300, C₂²×#7849977, C₂²×#7852137, …

GAP

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

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

C22×#63012 #72278967