D₇₈₄₇ #30628006

order	15694
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₈₄₇
unambiguous semidirect product	C₇⋊D₁₁₂₁
unambiguous semidirect product	C₁₉⋊D₄₁₃
unambiguous semidirect product	C₅₉⋊D₁₃₃

Splittings

Fitting, socle, derived, Hall	C₇×C₁₉×C₅₉ ⋊ C₂
Hall	C₁₉×C₅₉ ⋊ D₇
Hall	C₇×C₅₉ ⋊ D₁₉
Hall	C₇×C₁₉ ⋊ D₅₉
Sylow	C₅₉ ⋊ D₁₃₃
Sylow	C₁₉ ⋊ D₄₁₃
Sylow	C₇ ⋊ D₁₁₂₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₉×C₅₉	C₂
LCS	G ▷ C₇×C₁₉×C₅₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 7847

A minimal faithful permutation representation — 2 generators on 85 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);(2,7)(3,6)(4,5)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,7,19,59|b^a=b^6,c^a=c^18,d^a=d^58

Conjugacy classes & character table

3925 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₉×D₁₃₃, D₁₉×D₄₁₃, D₇×D₁₁₂₁, …
maximal normal subgroup of	D₅₉×D₁₃₃, D₁₉×D₄₁₃, D₇×D₁₁₂₁, …
maximal quotient of	Dic₇₈₄₇, C₂×D₇₈₄₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15694, 8);

D7847 #30628006