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C₃×C₄×D₁₉ #90705

order	456
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

C₃×C₄×D₁₉

Splittings

Fitting	C₃×C₄×C₁₉ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂×C₃×C₄
Hall	C₃×C₁₉ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	D₁₉
commutator, hypocenter	C₁₉	C₂×C₃×C₄
Frattini	C₂	C₂×C₃×D₁₉
Fitting	C₃×C₄×C₁₉	C₂
socle	C₂×C₃×C₁₉	C₂²
prime radical	C₂×C₃×D₁₉	C₂
cosocle	C₂×C₁₉	C₂²×C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₃×C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 228

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊C₈×D₁₉, #11744815, C₃×C₈×D₁₉, …
maximal normal subgroup of	C₃⋊C₈×D₁₉, #11744815, C₃×C₈×D₁₉, …
maximal quotient of	C₃×C₈×D₁₉, C₃×#64515, C₃×C₄×Dic₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 25);