C₁₉×#7186 #18013145

order	4864
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7186

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₁₉
Frattini, cosocle	C₂³×C₈	C₂²×C₁₉
socle	C₂³×C₁₉	C₂²⋊C₈
prime radical	C₂⁴×C₁₉	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

326 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 63
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 2 generators on 63 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25)(26,30,31,28)(27,29)(32,33,36,41,37,42,51,59,38,43,52,60,35,40,49,57)(34,44,46,61,45,55,62,58,47,56,50,63,48,39,53,54);(20,21,22,23)(24,26)(25,28)(27,31)(29,30)(32,34)(33,39)(35,48)(36,50)(37,45)(38,47)(40,56)(41,58)(42,44)(43,55)(46,52)(49,62)(51,53)(54,60)(57,61)(59,63)

Conjugacy classes & character table

1444 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 174);