C₂²⋊C₉×C₁₀₉ #16840987

order	3924
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₀₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂²⋊C₉×C₁₀₉
unambiguous semidirect product	C₂²⋊(C₉×C₁₀₉)

Splittings

derived, Sylow	C₂² ⋊ C₉×C₁₀₉
Hall	C₂²×C₁₀₉ ⋊ C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₀₉	A₄
commutator, hypocenter	C₂²	C₉×C₁₀₉
Frattini	C₃	A₄×C₁₀₉
Fitting, socle, prime radical	C₂²×C₃×C₁₀₉	C₃
cosocle	C₂²×C₃	C₃×C₁₀₉
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₁₀₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 122
transitive degree	μ_t(G) = 1962

A minimal faithful permutation representation — 2 generators on 122 points:

(2,3,4)(5,6,8,7,9,11,10,12,13)(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);(1,4)(2,3)

Conjugacy classes & character table

1308 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×C₂²⋊C₉×C₁₀₉, C₁₀₉×C₂²⋊(C₇⋊C₉), C₇×C₂²⋊C₉×C₁₀₉
maximal normal subgroup of	C₅×C₂²⋊C₉×C₁₀₉, C₇×C₂²⋊C₉×C₁₀₉
maximal quotient of	C₅×C₂²⋊C₉×C₁₀₉, C₁₀₉×C₂²⋊(C₇⋊C₉), C₇×C₂²⋊C₉×C₁₀₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3924, 14);

C22⋊C9×C109 #16840987