C₂⁷×D₈ #72314419

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁶

Aliases

direct product

C₂⁷×D₈

Subgroups

	subgroup	G/H
center, socle	C₂⁸	D₄
commutator, Frattini, cosocle	C₄	C₂⁹
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁸ ◁ C₂⁷×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

24015976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 9 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁶×#442, C₂⁷×AGL(1,ℤ/8), C₂⁸×D₈
maximal normal subgroup of	C₂⁶×#442, C₂⁷×AGL(1,ℤ/8), C₂⁸×D₈
maximal quotient of	C₂⁷×#102, C₂⁸×D₈

GAP

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

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

C27×D8 #72314419