C₂×C₇×C₁₀₃⋊C₃ #16864484

order	4326
isoclinism	isoclinic class of C₁₀₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₇×C₁₀₃⋊C₃
unambiguous semidirect product	(C₂×C₁₀₃)⋊(C₃×C₇)

Splittings

Fitting, socle, Hall	C₂×C₇×C₁₀₃ ⋊ C₃
derived, Sylow	C₁₀₃ ⋊ C₂×C₃×C₇
Hall	C₇×C₁₀₃ ⋊ C₂×C₃
Hall	C₂×C₁₀₃ ⋊ C₃×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	C₁₀₃⋊C₃
commutator, cosocle, hypocenter	C₁₀₃	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₀₃	C₃
LCS	G ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 1442

A minimal faithful permutation representation — 2 generators on 112 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×ΓL(1,𝔽₈)×C₁₀₃⋊C₃, D₃×C₇×C₁₀₃⋊C₃, C₂×C₃×C₇×C₁₀₃⋊C₃, …
maximal normal subgroup of	C₂×ΓL(1,𝔽₈)×C₁₀₃⋊C₃, C₂×C₃×C₇×C₁₀₃⋊C₃, C₂×C₅×C₇×C₁₀₃⋊C₃, …
maximal quotient of	C₂×C₇×C₁₀₃⋊C₉, D₃×C₇×C₁₀₃⋊C₃, C₂×C₃×C₇×C₁₀₃⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4326, 2);

C2×C7×C103⋊C3 #16864484