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C₈×D₄×C₁₉ #11926110

order	1216
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

C₈×D₄×C₁₉

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₉	C₂²
commutator	C₂	C₂²×C₈×C₁₉
Frattini, cosocle	C₂×C₄	C₂³×C₁₉
socle	C₂²×C₁₉	C₂²×C₄
prime radical	C₂×D₄×C₁₉	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×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:

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

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₄⋊(C₁₉⋊C₁₆), C₈×#64525, C₈×#64526, …
maximal subgroup of	C₁₉×D₄⋊C₁₆, C₈×D₈×C₁₉, C₈×SD₁₆×C₁₉, …
maximal normal subgroup of	C₁₉×D₄⋊C₁₆, C₈×D₈×C₁₉, C₈×SD₁₆×C₁₉, …
maximal quotient of	C₈×D₈×C₁₉, C₈×SD₁₆×C₁₉, C₈×Q₁₆×C₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 115);