C₇×C₅⋊(C₁₉⋊D₁₉) #38760149

order	25270
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁₉

Aliases

direct product	C₇×C₅⋊(C₁₉⋊D₁₉)
unambiguous semidirect product	C₅⋊(C₇×C₁₉⋊D₁₉)
unambiguous semidirect product	C₁₉⋊(C₇×D₉₅)

Splittings

Fitting, socle, Hall	C₅×C₇×C₁₉² ⋊ C₂
derived, Hall	C₅×C₁₉² ⋊ C₂×C₇
Hall	C₇×C₁₉² ⋊ D₅
Sylow	C₁₉² ⋊ C₇×D₅
Hall	C₅×C₇ ⋊ C₁₉⋊D₁₉
Sylow	C₅ ⋊ C₇×C₁₉⋊D₁₉

Subgroups

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

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:

(1,2,3,4,5,6,7)(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);(8,9,11,12,10)(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

6328 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25270, 23);

C7×C5⋊(C19⋊D19) #38760149