C₅×Dic₁₉ #67410

order	380
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×Dic₁₉
unambiguous semidirect product	C₁₉⋊(C₄×C₅)

Splittings

derived, Sylow	C₁₉ ⋊ C₄×C₅
Hall	C₅×C₁₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₁₉
commutator, hypocenter	C₁₉	C₄×C₅
Frattini	C₂	C₅×D₁₉
Fitting, socle, prime radical	C₂×C₅×C₁₉	C₂
cosocle	C₂×C₁₉	C₂×C₅
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₅

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 380

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×Dic₁₉, #10625572, #10625575, …
maximal normal subgroup of	D₅×Dic₁₉, #10625572, #10625575, …
maximal quotient of	C₅×C₁₉⋊C₈, C₂×C₅×Dic₁₉, C₃×C₅×Dic₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(380, 2);

C5×Dic19 #67410

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₅×Dic₁₉ #67410