C₅²×D₇×D₁₉ #27210883

order	13300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅² · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₅²×D₇×D₁₉
unambiguous semidirect product	(D₇×C₁₉)⋊(C₂×C₅²)
unambiguous semidirect product	(C₇×D₁₉)⋊(C₂×C₅²)
unambiguous semidirect product	D₁₃₃⋊(C₂×C₅²)

Splittings

Fitting, socle, Hall	C₅²×C₇×C₁₉ ⋊ C₂²
derived, Hall	C₇×C₁₉ ⋊ C₂²×C₅²
Hall	C₅²×C₁₉ ⋊ C₂×D₇
Hall	C₅²×C₇ ⋊ C₂×D₁₉
Sylow	C₁₉ ⋊ C₂×C₅²×D₇
Sylow	C₇ ⋊ C₂×C₅²×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅²	D₇×D₁₉
commutator, cosocle, hypocenter	C₇×C₁₉	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₁₉	C₂²
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₅²

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 3325

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

Conjugacy classes & character table

1375 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 41);

C52×D7×D19 #27210883