C₁₀₃⋊C₃ #64576

order	309
isoclinism	isoclinic class of C₁₀₃⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₁₀₃⋊C₃

Splittings

Fitting, socle, derived, Sylow

C₁₀₃ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₀₃	C₃
LCS	G ▷ C₁₀₃
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 103

A minimal faithful permutation representation — 2 generators on 103 points:

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

Polycyclic presentation:

3,103|b^a=b^56

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₀₃⋊C₃, C₂×C₁₀₃⋊C₃, C₃×C₁₀₃⋊C₃, …
maximal normal subgroup of	D₁₀₃⋊C₃, C₂×C₁₀₃⋊C₃, C₃×C₁₀₃⋊C₃, …
maximal quotient of	C₂×C₁₀₃⋊C₃, C₁₀₃⋊C₉, C₃×C₁₀₃⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(309, 1);

C103⋊C3 #64576