C₂⁴×AGL(1,ℤ/16) #72286515

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂⁴×AGL(1,ℤ/16)

Subgroups

	subgroup	G/H
center, socle	C₂⁵	#442
commutator	C₈	C₂⁶×C₄
Frattini, cosocle	C₂×C₈	C₂⁷
prime radical	C₂⁴×#509	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄ ◁ C₂⁴×#97 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

183704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

(1,2);(3,4);(9,10,14,18,23,17,12,22,15,20,24,16,13,19,21,11);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,17)(15,20)(18,21)(19,23)(22,24);(5,6)(9,11,21,19,13,16,24,20,15,22,12,17,23,18,14,10);(9,11,13,18,15,22,23,16)(10,14,17,21,20,24,19,12);(1,2)(3,4)(5,6)(9,12,23,24,15,21,13,14)(10,16,19,22,20,18,17,11)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#495725, C₂×#72257081, C₂²×D₄×AGL(1,ℤ/16), …
maximal normal subgroup of	C₂³×#495725, C₂×#72257081, C₂²×D₄×AGL(1,ℤ/16), …
maximal quotient of	C₂⁴×#12067, C₂³×#495725, C₂²×D₄×AGL(1,ℤ/16), …

GAP

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

gap> G := Group((1,2), (3,4), (9,10,14,18,23,17,12,22,15,20,24,16,13,19,21,11), (1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,17)(15,20)(18,21)(19,23)(22,24), (5,6)(9,11,21,19,13,16,24,20,15,22,12,17,23,18,14,10), (9,11,13,18,15,22,23,16)(10,14,17,21,20,24,19,12), (1,2)(3,4)(5,6)(9,12,23,24,15,21,13,14)(10,16,19,22,20,18,17,11));

C24×AGL(1,ℤ/16) #72286515