Finite Group Databasebeta

← Prev (#10597402) Next (#10597404) →

C₁₉×#117 #10597403

order	608
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

Subgroups

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

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 304

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23)(24,25)(26,29)(27,28)(30,31);(25,28)(29,31);(24,26,27,30)(25,29,28,31)

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11926300, #11926303, #11926314, …
maximal subgroup of	C₁₉×H(3,ℤ/4), C₁₉×#339, C₁₉×#344, …
maximal normal subgroup of	C₁₉×H(3,ℤ/4), C₁₉×#339, C₁₉×#344, …
maximal quotient of	C₁₉×#376, C₄×C₂²⋊C₄×C₁₉, C₄×C₄⋊C₄×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(608, 152);