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C₁₃×C₃⋊#1042 #16839174

order	3900
isoclinism	isoclinic class of C₃⋊#1042
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×C₃⋊#1042
unambiguous semidirect product	C₃⋊(C₁₃×#1042)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₁₃ ⋊ C₄
derived, Hall	C₃×C₅² ⋊ C₄×C₁₃
Hall	C₅²×C₁₃ ⋊ Dic₃
Hall	C₃×C₁₃ ⋊ #1042
Sylow	C₅² ⋊ Dic₃×C₁₃
Sylow	C₃ ⋊ C₁₃×#1042

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃	C₃⋊#1042
commutator, hypocenter	C₃×C₅²	C₄×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₃	C₄
prime radical	C₃×C₅×D₅×C₁₃	C₂
cosocle	C₃×C₅×D₅	C₂×C₁₃
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₁₃

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 780

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₃×C₉⋊#1042, C₁₃⋊C₃×C₃⋊#1042, C₃×C₁₃×C₃⋊#1042, …
maximal normal subgroup of	C₁₃⋊C₃×C₃⋊#1042, C₃×C₁₃×C₃⋊#1042, C₇×C₁₃×C₃⋊#1042
maximal quotient of	C₁₃×C₉⋊#1042, C₃×C₁₃×C₃⋊#1042, C₁₃×C₃⋊(C₃⋊#1042), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3900, 182);