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C₂×C₁₉×#117 #11926190

order	1216
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

C₂×C₁₉×#117

Subgroups

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

330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 4 generators on 33 points:

(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33);(3,4,5,6)(7,8)(9,12)(10,11)(13,14);(8,11)(12,14);(7,9,10,13)(8,12,11,14)

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14561949, #14561969, #14563332, …
maximal subgroup of	C₁₉×#1458, C₁₉×#1464, C₁₉×#1729, …
maximal normal subgroup of	C₁₉×#1458, C₁₉×#1464, C₁₉×#1729, …
maximal quotient of	C₂×C₁₉×#376, C₂×C₄×C₂²⋊C₄×C₁₉, C₂×C₄×C₄⋊C₄×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 195);