C₁₉×#12897 #18018856

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12897

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#117
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	#432
prime radical	C₂³×C₁₉	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

278 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 1216

A minimal faithful permutation representation — 3 generators on 59 points:

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

Conjugacy classes & character table

1216 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5885);