C₃×C₇⋊(C₁₉⋊D₁₉) #29503161

order	15162
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁₉

Aliases

direct product

C₃×C₇⋊(C₁₉⋊D₁₉)

Splittings

Fitting, socle, Hall	C₃×C₇×C₁₉² ⋊ C₂
derived, Hall	C₇×C₁₉² ⋊ C₂×C₃
Hall	C₃×C₁₉² ⋊ D₇
Sylow	C₁₉² ⋊ C₃×D₇
Hall	C₃×C₇ ⋊ C₁₉⋊D₁₉
Sylow	C₇ ⋊ C₃×C₁₉⋊D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₇⋊(C₁₉⋊D₁₉)
commutator, cosocle, hypocenter	C₇×C₁₉²	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₉²	C₂
LCS	G ▷ C₇×C₁₉²
UCS	C₁ ◁ C₃

176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 7581

A minimal faithful permutation representation — 3 generators on 48 points:

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

Conjugacy classes & character table

3795 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₁₉×D₁₃₃, C₃×D₇×C₁₉⋊D₁₉, D₅₇⋊D₁₃₃, …
maximal normal subgroup of	C₃×D₁₉×D₁₃₃, C₃×D₇×C₁₉⋊D₁₉, D₅₇⋊D₁₃₃, …
maximal quotient of	C₃×C₇⋊(C₁₉⋊Dic₁₉), C₂×C₃×C₇⋊(C₁₉⋊D₁₉)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15162, 96);

C3×C7⋊(C19⋊D19) #29503161