C₂×C₇⋊C₉×C₁₀₃ #27188508

order	12978
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂×C₉×C₁₀₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₇⋊C₉×C₁₀₃
unambiguous semidirect product	(C₂×C₇)⋊(C₉×C₁₀₃)

Splittings

derived, Sylow	C₇ ⋊ C₂×C₉×C₁₀₃
Hall	C₂×C₇×C₁₀₃ ⋊ C₉
Hall	C₇×C₁₀₃ ⋊ C₂×C₉
Hall	C₂×C₇ ⋊ C₉×C₁₀₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₀₃	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₂×C₉×C₁₀₃
Frattini	C₃	C₂×ΓL(1,𝔽₈)×C₁₀₃
Fitting, socle, prime radical	C₂×C₃×C₇×C₁₀₃	C₃
cosocle	C₃×C₇	C₂×C₃×C₁₀₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₁₀₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 12978

A minimal faithful permutation representation — 2 generators on 121 points:

(2,3,4)(5,7,6)(8,15,14,10,9,16,13,12,11)(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);(1,2,3,5,4,6,7)

Conjugacy classes & character table

3090 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×C₇⋊C₉×C₁₀₃, C₇⋊C₉×Dic₁₀₃, C₁₀₃×Dic₇⋊C₉, …
maximal normal subgroup of	C₄×C₇⋊C₉×C₁₀₃, C₇⋊C₉×Dic₁₀₃, C₁₀₃×Dic₇⋊C₉, …
maximal quotient of	C₄×C₇⋊C₉×C₁₀₃, C₂²×C₇⋊C₉×C₁₀₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12978, 3);

C2×C7⋊C9×C103 #27188508