A₄×C₁₆₉ #14523120

order	2028
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₆₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₁₆₉
unambiguous semidirect product	C₂²⋊(C₃×C₁₆₉)

Splittings

Fitting, Hall	C₂²×C₁₆₉ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₁₆₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₆₉	A₄
commutator, hypocenter	C₂²	C₃×C₁₆₉
Frattini	C₁₃	A₄×C₁₃
Fitting	C₂²×C₁₆₉	C₃
socle, cosocle	C₂²×C₁₃	C₃×C₁₃
prime radical	A₄×C₁₃	C₁₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₁₆₉

15 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 173
transitive degree	μ_t(G) = 676

A minimal faithful permutation representation — 2 generators on 173 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173);(1,2,3)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×A₄×C₁₆₉, C₂²⋊ΓL(1,𝔽₈)×C₁₆₉, C₁₁×A₄×C₁₆₉, …
maximal normal subgroup of	C₇×A₄×C₁₆₉, C₁₁×A₄×C₁₆₉, A₄×C₁₇×C₁₆₉, …
maximal quotient of	C₇×A₄×C₁₆₉, C₂²⋊ΓL(1,𝔽₈)×C₁₆₉, C₁₁×A₄×C₁₆₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2028, 14);

A4×C169 #14523120