C₅×C₇×D₅×D₁₉ #27210888

order	13300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×C₇×D₅×D₁₉
unambiguous semidirect product	(D₅×C₁₉)⋊(C₂×C₅×C₇)
unambiguous semidirect product	(C₅×D₁₉)⋊(C₂×C₅×C₇)
unambiguous semidirect product	D₉₅⋊(C₂×C₅×C₇)

Splittings

Fitting, socle, Hall	C₅²×C₇×C₁₉ ⋊ C₂²
derived	C₅×C₁₉ ⋊ C₂²×C₅×C₇
Hall	C₅²×C₁₉ ⋊ C₂²×C₇
Hall	C₅²×C₇ ⋊ C₂×D₁₉
Hall	C₇×C₁₉ ⋊ C₂×C₅×D₅
Sylow	C₅² ⋊ C₂×C₇×D₁₉
Sylow	C₁₉ ⋊ C₂×C₅×C₇×D₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₇	D₅×D₁₉
commutator, cosocle, hypocenter	C₅×C₁₉	C₂²×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₁₉	C₂²
LCS	G ▷ C₅×C₁₉
UCS	C₁ ◁ C₅×C₇

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 1330

A minimal faithful permutation representation — 2 generators on 36 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21,22,23,24)(25,26,28,29,27)(30,31,32,33,34,35,36);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(26,27)(28,29)

Conjugacy classes & character table

1540 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 46);

C5×C7×D5×D19 #27210888