Q₈×Q₁₆×C₁₉ #14561239

order	2432
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

Q₈×Q₁₆×C₁₉

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×Q₈×C₁₉ ◁ G

336 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 43
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 4 generators on 43 points:

(1,2,3,4)(5,8,7,6)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(1,5,3,7)(2,6,4,8);(9,10,11,12,13,14,15,16)(17,24,23,22,21,20,19,18);(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)

Conjugacy classes & character table

665 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₉×#12354, C₁₉×#12358, C₁₉×#12406, …
maximal normal subgroup of	C₁₉×#12354, C₁₉×#12358, C₁₉×#12406, …
maximal quotient of	Q₈×C₁₉×#103, C₄⋊C₄×Q₁₆×C₁₉, Q₈×C₁₉×#107, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2432, 2114);