C₁₉×C₇⋊#1039 #27210880

order	13300
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₉ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₁₉×C₇⋊#1039
unambiguous semidirect product	C₇⋊(C₁₉×#1039)

Splittings

derived, Hall	C₅²×C₇ ⋊ C₄×C₁₉
Hall	C₅²×C₇×C₁₉ ⋊ C₄
Hall	C₅²×C₁₉ ⋊ Dic₇
Hall	C₇×C₁₉ ⋊ #1039
Sylow	C₅² ⋊ C₁₉×Dic₇
Sylow	C₇ ⋊ C₁₉×#1039

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₉	C₅⋊D₃₅
commutator, hypocenter	C₅²×C₇	C₄×C₁₉
Frattini	C₂	C₁₉×C₅⋊D₃₅
Fitting, socle, prime radical	C₂×C₅²×C₇×C₁₉	C₂
cosocle	C₂×C₅²×C₇	C₂×C₁₉
LCS	G ▷ C₅²×C₇
UCS	C₁ ◁ C₂×C₁₉

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 13300

A minimal faithful permutation representation — 3 generators on 40 points:

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

Conjugacy classes & character table

3382 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 38);

C19×C7⋊#1039 #27210880