C₇×C₃⋊D₃×C₁₀₃ #27188570

order	12978
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₁₀₃ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₇×C₃⋊D₃×C₁₀₃
unambiguous semidirect product	C₃⋊(D₃×C₇×C₁₀₃)

Splittings

Fitting, socle, Hall	C₃²×C₇×C₁₀₃ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₇×C₁₀₃
Hall	C₃²×C₁₀₃ ⋊ C₂×C₇
Hall	C₃²×C₇ ⋊ C₂×C₁₀₃

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₁₀₃	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂×C₇×C₁₀₃
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₁₀₃	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₇×C₁₀₃

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 116
transitive degree	μ_t(G) = 6489

A minimal faithful permutation representation — 3 generators on 116 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103)(105,106)(108,109)(110,111,112,113,114,115,116);(107,108,109);(104,105,106)

Conjugacy classes & character table

4326 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×#157×C₁₀₃, C₇×C₁₀₃⋊#157, C₁₀₃×C₇⋊#157, …
maximal normal subgroup of	C₇×#157×C₁₀₃, C₇×C₁₀₃⋊#157, C₁₀₃×C₇⋊#157, …
maximal quotient of	C₇×C₃⋊Dic₃×C₁₀₃, C₂×C₇×C₃⋊D₃×C₁₀₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12978, 65);

C7×C3⋊D3×C103 #27188570