C₁₉×Q₈⋊C₃₂ #18013351

order	4864
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₉×Q₈⋊C₃₂
unambiguous semidirect product	Q₈⋊(C₁₉×C₃₂)

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₁₉	D₄
commutator	C₄	C₂×C₁₉×C₃₂
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₉
socle, prime radical	C₂²×C₁₉	C₂²⋊C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₉ ◁ C₄×C₁₆×C₁₉ ◁ G

118 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 59
transitive degree	μ_t(G) = 4864

A minimal faithful permutation representation — 2 generators on 59 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,24)(22,26)(25,27)(28,29,30,34,31,35,38,44,32,36,39,45,41,47,50,54,33,37,40,46,42,48,51,55,43,49,52,56,53,57,58,59);(20,21,23,25)(22,27,26,24)

Conjugacy classes & character table

2128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 380);