C₁₃×C₂²⋊(C₅²⋊C₃) #16839196

order	3900
isoclinism	isoclinic class of C₂²⋊(C₅²⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₃ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₁₃×C₂²⋊(C₅²⋊C₃)
unambiguous semidirect product	C₂²⋊(C₁₃×C₅²⋊C₃)
unambiguous semidirect product	C₅²⋊(A₄×C₁₃)

Splittings

Fitting, socle, Hall	C₂²×C₅²×C₁₃ ⋊ C₃
derived, Hall	C₂²×C₅² ⋊ C₃×C₁₃
Hall	C₅²×C₁₃ ⋊ A₄
Hall	C₂²×C₁₃ ⋊ C₅²⋊C₃
Sylow	C₅² ⋊ A₄×C₁₃
Sylow	C₂² ⋊ C₁₃×C₅²⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃	C₂²⋊(C₅²⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₅²	C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₁₃	C₃
LCS	G ▷ C₂²×C₅²
UCS	C₁ ◁ C₁₃

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 390

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₃⋊C₃×C₂²⋊(C₅²⋊C₃), A₄×C₁₃×C₅²⋊C₃, (C₂²⋊(C₁₃⋊C₃))⋊(C₅²⋊C₃), …
maximal normal subgroup of	C₁₃⋊C₃×C₂²⋊(C₅²⋊C₃), A₄×C₁₃×C₅²⋊C₃, (C₂²⋊(C₁₃⋊C₃))⋊(C₅²⋊C₃), …
maximal quotient of	C₁₃×C₂²⋊(C₅²⋊C₉), C₃×C₁₃×C₂²⋊(C₅²⋊C₃), C₇×C₁₃×C₂²⋊(C₅²⋊C₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3900, 204);

C13×C22⋊(C52⋊C3) #16839196