C₂⁷×AGL(1,ℤ/8) #72314428

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂¹⁰ · Schur multiplier (H₂) = C₂⁴⁴

Aliases

direct product

C₂⁷×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center, socle	C₂⁸	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂¹⁰
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁸ ◁ C₂⁸×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

565896955 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 10 generators on 22 points:

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

Conjugacy classes & character table

1408 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₂⁷×#442

GAP

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

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

C27×AGL(1,ℤ/8) #72314428