C₇×C₁₉⋊#1039 #27210879

order	13300
automorphism	?
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₅

Aliases

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

Splittings

derived, Hall	C₅²×C₁₉ ⋊ C₄×C₇
Hall	C₅²×C₇×C₁₉ ⋊ C₄
Hall	C₅²×C₇ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ #1039
Sylow	C₅² ⋊ C₇×Dic₁₉
Sylow	C₁₉ ⋊ C₇×#1039

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	C₅⋊D₉₅
commutator, hypocenter	C₅²×C₁₉	C₄×C₇
Frattini	C₂	C₇×C₅⋊D₉₅
Fitting, socle, prime radical	C₂×C₅²×C₇×C₁₉	C₂
cosocle	C₂×C₅²×C₁₉	C₂×C₇
LCS	G ▷ C₅²×C₁₉
UCS	C₁ ◁ C₂×C₇

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

Conjugacy classes & character table

3346 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 37);

C7×C19⋊#1039 #27210879