C₅²×D₅₇ #15668147

order	2850
isoclinism	isoclinic class of D₅₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₅² · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²×D₅₇
unambiguous semidirect product	C₃⋊(C₅²×D₁₉)
unambiguous semidirect product	C₁₉⋊(C₅²×D₃)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₁₉ ⋊ C₂
derived, Hall	C₃×C₁₉ ⋊ C₂×C₅²
Hall	C₅²×C₁₉ ⋊ D₃
Hall	C₃×C₅² ⋊ D₁₉
Sylow	C₁₉ ⋊ C₅²×D₃
Sylow	C₃ ⋊ C₅²×D₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅²	D₅₇
commutator, cosocle, hypocenter	C₃×C₁₉	C₂×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₉	C₂
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₅²

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 1425

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅²×D₃₉₉, C₅²×C₇×D₅₇, C₅×C₃⋊(C₁₉⋊AGL(1,𝔽₁₁)), …
maximal normal subgroup of	C₅²×C₇×D₅₇, C₅²×C₁₁×D₅₇, C₅²×C₁₃×D₅₇
maximal quotient of	C₅²×D₃₉₉, C₅²×C₇×D₅₇, C₅×C₃⋊(C₁₉⋊AGL(1,𝔽₁₁)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2850, 36);

C52×D57 #15668147