C₂²×#648367 #72321316

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹×C₄

Aliases

direct product

C₂²×#648367

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

1174440 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#120×#121
maximal normal subgroup of	C₂²×#120×#121
maximal quotient of	C₂²×#72278310

GAP

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

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

C22×#648367 #72321316