C₅×C₇⋊(C₁₉⋊D₁₉) #38760146

order	25270
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁₉

Aliases

direct product	C₅×C₇⋊(C₁₉⋊D₁₉)
unambiguous semidirect product	C₇⋊(C₅×C₁₉⋊D₁₉)
unambiguous semidirect product	C₁₉⋊(C₅×D₁₃₃)

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₉² ⋊ C₂
derived, Hall	C₇×C₁₉² ⋊ C₂×C₅
Hall	C₅×C₁₉² ⋊ D₇
Sylow	C₁₉² ⋊ C₅×D₇
Hall	C₅×C₇ ⋊ C₁₉⋊D₁₉
Sylow	C₇ ⋊ C₅×C₁₉⋊D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₇⋊(C₁₉⋊D₁₉)
commutator, cosocle, hypocenter	C₇×C₁₉²	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₉²	C₂
LCS	G ▷ C₇×C₁₉²
UCS	C₁ ◁ C₅

176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 12635

A minimal faithful permutation representation — 3 generators on 50 points:

(1,2,3,4,5)(7,8)(9,10)(11,12)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50);(6,7,9,11,12,10,8)(32,33,35,37,39,41,43,45,47,49,50,48,46,44,42,40,38,36,34);(13,14,16,18,20,22,24,26,28,30,31,29,27,25,23,21,19,17,15)

Conjugacy classes & character table

6325 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25270, 20);

C5×C7⋊(C19⋊D19) #38760146