C₂⁷×SD₁₆ #72314423

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

Aliases

direct product

C₂⁷×SD₁₆

Subgroups

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

20276840 conjugacy classes of subgroups; only the 9059432 normal subgroups are shown.

normal subgroups only

This group has 20276840 conjugacy classes of subgroups — too many to draw. Only its 9059432 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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);(11,12);(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)(15,16,18,20)(17,22,21,19);(13,14)

Conjugacy classes & character table

896 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	C₂⁷×#102, C₂⁷×#106

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), (11,12), (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)(15,16,18,20)(17,22,21,19), (13,14));