D₇₁₈₉ #28385487

order	14378
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₁₈₉
unambiguous semidirect product	C₇⋊D₁₀₂₇
unambiguous semidirect product	C₁₃⋊D₅₅₃
unambiguous semidirect product	C₇₉⋊D₉₁

Splittings

Fitting, socle, derived, Hall	C₇×C₁₃×C₇₉ ⋊ C₂
Hall	C₁₃×C₇₉ ⋊ D₇
Hall	C₇×C₇₉ ⋊ D₁₃
Hall	C₇×C₁₃ ⋊ D₇₉
Sylow	C₇₉ ⋊ D₉₁
Sylow	C₁₃ ⋊ D₅₅₃
Sylow	C₇ ⋊ D₁₀₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₇₉	C₂
LCS	G ▷ C₇×C₁₃×C₇₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 7189

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

Polycyclic presentation:

2,7,13,79|b^a=b^6,c^a=c^12,d^a=d^78

Conjugacy classes & character table

3596 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₉×D₉₁, D₁₃×D₅₅₃, D₇×D₁₀₂₇, …
maximal normal subgroup of	D₇₉×D₉₁, D₁₃×D₅₅₃, D₇×D₁₀₂₇, …
maximal quotient of	Dic₇₁₈₉, C₂×D₇₁₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14378, 9);

D7189 #28385487