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C₄×D₈×C₁₉ #11926113

order	1216
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

C₄×D₈×C₁₉

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	D₄
commutator	C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×D₄
prime radical	C₂×D₈×C₁₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₄²×C₁₉ ◁ G

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 608

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

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

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14561596, #14561602, D₈×C₁₉⋊C₈, …
maximal subgroup of	C₁₉×#1331, C₁₉×#1333, C₈×D₈×C₁₉, …
maximal normal subgroup of	C₁₉×#1331, C₁₉×#1333, C₈×D₈×C₁₉, …
maximal quotient of	C₈×D₈×C₁₉, C₁₉×#1581, C₁₉×#1589, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 118);