C₁₉×Dic₅₅ #16862016

order	4180
isoclinism	isoclinic class of D₅₅
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×Dic₅₅
unambiguous semidirect product	C₁₁⋊(C₁₉×Dic₅)

Splittings

derived, Hall	C₅×C₁₁ ⋊ C₄×C₁₉
Hall	C₅×C₁₁×C₁₉ ⋊ C₄
Hall	C₁₁×C₁₉ ⋊ Dic₅
Hall	C₅×C₁₉ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ C₁₉×Dic₅
Sylow	C₅ ⋊ C₁₉×Dic₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₉	D₅₅
commutator, hypocenter	C₅×C₁₁	C₄×C₁₉
Frattini	C₂	C₁₉×D₅₅
Fitting, socle, prime radical	C₂×C₅×C₁₁×C₁₉	C₂
cosocle	C₂×C₅×C₁₁	C₂×C₁₉
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₂×C₁₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 4180

A minimal faithful permutation representation — 2 generators on 39 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)(29,30,31,32,33,34,35,36,37,38,39);(20,22,21,23)(25,28)(26,27)(30,39)(31,38)(32,37)(33,36)(34,35)

Conjugacy classes & character table

1102 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₁₉×Dic₅₅, C₁₉×Dic₁₆₅, C₁₉⋊C₃×Dic₅₅, …
maximal normal subgroup of	C₃×C₁₉×Dic₅₅, C₁₉⋊C₃×Dic₅₅, C₁₉×(C₁₁⋊C₅)⋊Dic₅, …
maximal quotient of	C₃×C₁₉×Dic₅₅, C₁₉×Dic₁₆₅, C₁₉×Dic₂₇₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4180, 8);

C19×Dic55 #16862016