A₄×C₁₁₃ #13062619

order	1356
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₁₁₃
unambiguous semidirect product	C₂²⋊(C₃×C₁₁₃)
unambiguous semidirect product	(C₂²×C₁₁₃)⋊C₃

Splittings

Fitting, socle, Hall	C₂²×C₁₁₃ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₁₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁₃	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₁₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₁₁₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₁₁₃

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 452

A minimal faithful permutation representation — 2 generators on 117 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(1,2,3)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×A₄×C₁₁₃, C₁₁×A₄×C₁₁₃, C₁₁₃×C₂²⋊(C₁₃⋊C₃), …
maximal normal subgroup of	C₃×A₄×C₁₁₃, C₁₁×A₄×C₁₁₃, A₄×C₁₃×C₁₁₃, …
maximal quotient of	C₂²⋊C₉×C₁₁₃, C₃×A₄×C₁₁₃, C₁₁×A₄×C₁₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1356, 8);

A4×C113 #13062619