C₁₉×#7325 #18013284

order	4864
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₉×#7325

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#342
commutator	C₂⁴	C₄²×C₁₉
Frattini, cosocle	C₂×#117	C₂²×C₁₉
prime radical	C₂⁴×C₁₉	C₄²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

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

normal subgroups only

This group has 244 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) = 67
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 67 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,28,29,25)(23,27)(24,26)(30,34,35,32)(31,33)(36,37,40,44,41,45,53,60)(38,46,48,61,49,57,62,66)(39,43,51,58,52,59,64,67)(42,54,55,50,56,63,65,47);(20,22,23,30,24,29,31,35)(21,25,27,32,26,28,33,34)(36,38)(37,42)(39,50,52,47)(40,48)(41,49)(43,57,59,46)(44,55)(45,56)(51,63,64,54)(53,62)(58,66,67,61)(60,65)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 313);