A₄×C₁₀₇ #13046207

order	1284
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₀₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₁₀₇
unambiguous semidirect product	C₂²⋊(C₃×C₁₀₇)
unambiguous semidirect product	(C₂²×C₁₀₇)⋊C₃

Splittings

Fitting, socle, Hall	C₂²×C₁₀₇ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₁₀₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁₀₇	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₁₀₇
Frattini	C₁	G
Fitting, socle	C₂²×C₁₀₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₁₀₇

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 428

A minimal faithful permutation representation — 2 generators on 111 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);(1,2,3)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×A₄×C₁₀₇, C₁₁×A₄×C₁₀₇, C₁₀₇×C₂²⋊(C₁₃⋊C₃), …
maximal normal subgroup of	C₃×A₄×C₁₀₇, C₁₁×A₄×C₁₀₇, A₄×C₁₃×C₁₀₇, …
maximal quotient of	C₂²⋊C₉×C₁₀₇, C₃×A₄×C₁₀₇, C₁₁×A₄×C₁₀₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1284, 6);

A4×C107 #13046207