C₂²×#62916 #72299722

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

Aliases

direct product

C₂²×#62916

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

508917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#10586570
maximal normal subgroup of	C₂²×#10586570
maximal quotient of	C₂²×#7880638, C₂²×#7688517, C₂²×#7989735

GAP

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

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

C22×#62916 #72299722