#157×C₁₀₉ #16841024

order	3924
isoclinism	isoclinic class of #157
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₀₉ · Schur multiplier (H₂) = C₃

Aliases

direct product

#157×C₁₀₉

Splittings

Fitting, socle, Hall	C₃²×C₁₀₉ ⋊ C₄
derived, Sylow	C₃² ⋊ C₄×C₁₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₀₉	#157
commutator, hypocenter	C₃²	C₄×C₁₀₉
Frattini	C₁	G
Fitting, socle	C₃²×C₁₀₉	C₄
prime radical	C₃⋊D₃×C₁₀₉	C₂
cosocle	C₃⋊D₃	C₂×C₁₀₉
LCS	G ▷ C₃²
UCS	C₁ ◁ C₁₀₉

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 654

A minimal faithful permutation representation — 2 generators on 115 points:

(1,2)(3,5,6,4)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,3,6)(2,5,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×#157×C₁₀₉, C₁₀₉×#4380, C₁₀₉×#4379, …
maximal normal subgroup of	C₅×#157×C₁₀₉, C₇×#157×C₁₀₉
maximal quotient of	C₅×#157×C₁₀₉, C₁₀₉×#4380, C₁₀₉×#4379, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3924, 51);

#157×C109 #16841024