C₂⁴×#3484 #72289437

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁵

Aliases

direct product

C₂⁴×#3484

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

21033800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 9 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7684106, C₂³×#7630939, C₂⁴×#63011, …
maximal normal subgroup of	C₂³×#7684106, C₂³×#7630939, C₂⁴×#63011, …
maximal quotient of	C₂³×#7684106, C₂⁴×D₄×#120, C₂³×#7630939, …
automorphism group of	#8411, #8863, #8878, …

GAP

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

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

C24×#3484 #72289437