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D₄×D₁₉ #64542

order	304
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

D₄×D₁₉

Splittings

Fitting	D₄×C₁₉ ⋊ C₂
Sylow	C₁₉ ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₁₉
commutator, socle, cosocle	C₂×C₁₉	C₂³
Fitting	D₄×C₁₉	C₂
hypercenter	D₄	D₁₉
hypocenter	C₁₉	C₂×D₄
LCS	G ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂ ◁ D₄

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 76

A minimal faithful permutation representation — 3 generators on 23 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(21,23);(20,21)(22,23)

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₈×D₁₉, #10597357, SD₁₆×D₁₉, …
maximal normal subgroup of	D₈×D₁₉, #10597357, SD₁₆×D₁₉, …
maximal quotient of	#10597324, C₂²⋊C₄×D₁₉, (C₂²⋊D₁₉)⋊C₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(304, 31);