C₂²×#63040 #72301085

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

Aliases

direct product

C₂²×#63040

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

539577 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:

(13,14,17,18)(15,16,19,20);(7,11)(10,12)(15,19)(16,20);(5,6)(7,12)(8,9)(10,11)(13,15,17,19)(14,16,18,20);(6,9)(7,11)(14,18)(16,20);(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,19)(17,18);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,19,18,15);(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×#10586740
maximal normal subgroup of	C₂²×#10586740
maximal quotient of	C₂²×#7688517

GAP

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

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

C22×#63040 #72301085