Dic₇×C₁₀₉ #15690232

order	3052
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₀₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₇×C₁₀₉
unambiguous semidirect product	C₇⋊(C₄×C₁₀₉)

Splittings

derived, Sylow	C₇ ⋊ C₄×C₁₀₉
Hall	C₇×C₁₀₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₀₉	D₇
commutator, hypocenter	C₇	C₄×C₁₀₉
Frattini	C₂	D₇×C₁₀₉
Fitting, socle, prime radical	C₂×C₇×C₁₀₉	C₂
cosocle	C₂×C₇	C₂×C₁₀₉
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₀₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 3052

A minimal faithful permutation representation — 2 generators on 120 points:

(1,2)(3,4)(5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120);(1,3,2,4)(6,11)(7,10)(8,9)

Conjugacy classes & character table

1090 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×Dic₇×C₁₀₉, C₁₀₉×Dic₃₅, C₁₀₉×C₇⋊AGL(1,𝔽₅), …
maximal normal subgroup of	C₅×Dic₇×C₁₀₉, C₇×Dic₇×C₁₀₉, C₁₁×Dic₇×C₁₀₉
maximal quotient of	C₅×Dic₇×C₁₀₉, C₁₀₉×Dic₃₅, C₁₀₉×C₇⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3052, 2);

Dic7×C109 #15690232