C₂⁷×Q₈ #72314422

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁵

Aliases

direct product

C₂⁷×Q₈

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁸	C₂²
commutator, Frattini, cosocle	C₂	C₂⁹
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁸ ◁ G
LECS	G ▷ C₂ ▷ C₁

8671445 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);(11,12)(13,14);(11,13)(12,14);(7,8)(9,10);(7,9)(8,10);(3,4)(5,6);(15,16,18,20)(17,21,22,19);(3,5)(4,6);(15,17,18,22)(16,19,20,21)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁷×SD₁₆, C₂⁷×#41
maximal normal subgroup of	C₂⁷×SD₁₆, C₂⁷×#41
maximal quotient of	C₂⁷×C₄⋊C₄

GAP

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

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

C27×Q8 #72314422