C₂²×#649052 #72320571

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

Aliases

direct product

C₂²×#649052

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

1345440 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);(6,8)(9,10,11,12);(5,6)(7,8)(9,10)(11,12)(17,18,20,19);(14,16)(18,19);(1,2)(3,4);(6,8)(10,12)(13,14,15,16)(17,19,20,18);(5,6)(7,8)(9,10)(11,12)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#120×#127
maximal normal subgroup of	C₂²×#120×#127
maximal quotient of	C₂²×#72278348, C₂²×#72278330

GAP

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

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

C22×#649052 #72320571