C₂²×#647714 #72289309

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂²×#647714

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6045 conjugacy classes of subgroups; only the 56600 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72219693, C₂²×#120×#121, C₂²×D₄×#2403
maximal normal subgroup of	C₂²×#72219693, C₂²×#120×#121, C₂²×D₄×#2403
maximal quotient of	C₂²×#72233926

GAP

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

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

C22×#647714 #72289309