C₅²×C₁₃×D₁₉ #27134709

order	12350
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅²×C₁₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²×C₁₃×D₁₉
unambiguous semidirect product	C₁₉⋊(C₂×C₅²×C₁₃)

Splittings

Fitting, socle, Hall	C₅²×C₁₃×C₁₉ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂×C₅²×C₁₃
Hall	C₅²×C₁₉ ⋊ C₂×C₁₃
Hall	C₁₃×C₁₉ ⋊ C₂×C₅²

Subgroups

	subgroup	G/H
center, hypercenter	C₅²×C₁₃	D₁₉
commutator, cosocle, hypocenter	C₁₉	C₂×C₅²×C₁₃
Frattini	C₁	G
Fitting, socle	C₅²×C₁₃×C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₅²×C₁₃

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 6175

A minimal faithful permutation representation — 2 generators on 42 points:

(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)(38,39,40,41,42);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(33,34,35,36,37)

Conjugacy classes & character table

3575 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅²×D₁₃×D₁₉, C₅×D₅×C₁₃×D₁₉, C₅×D₁₉×D₆₅, …
maximal normal subgroup of	C₅²×D₁₃×D₁₉, C₅×D₅×C₁₃×D₁₉, C₅×D₁₉×D₆₅, …
maximal quotient of	C₅²×C₁₃×Dic₁₉, C₂×C₅²×C₁₃×D₁₉, C₃×C₅²×C₁₃×D₁₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12350, 20);

C52×C13×D19 #27134709