C₂⁶×AGL(1,ℤ/8) #72313896

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁵

Aliases

direct product

C₂⁶×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁹
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ C₂⁷×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

20276840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁷×AGL(1,ℤ/8)
maximal normal subgroup of	C₂⁷×AGL(1,ℤ/8)
maximal quotient of	C₂⁶×#442, C₂⁷×AGL(1,ℤ/8)

GAP

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

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

C26×AGL(1,ℤ/8) #72313896