D₉₃₃₁ #32916254

order	18662
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₃₃₁
unambiguous semidirect product	C₇⋊D₁₃₃₃
unambiguous semidirect product	C₃₁⋊D₃₀₁
unambiguous semidirect product	C₄₃⋊D₂₁₇

Splittings

Fitting, socle, derived, Hall	C₇×C₃₁×C₄₃ ⋊ C₂
Hall	C₃₁×C₄₃ ⋊ D₇
Hall	C₇×C₄₃ ⋊ D₃₁
Hall	C₇×C₃₁ ⋊ D₄₃
Sylow	C₄₃ ⋊ D₂₁₇
Sylow	C₃₁ ⋊ D₃₀₁
Sylow	C₇ ⋊ D₁₃₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₃₁×C₄₃	C₂
LCS	G ▷ C₇×C₃₁×C₄₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 9331

A minimal faithful permutation representation — 2 generators on 81 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);(2,7)(3,6)(4,5)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(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:

2,7,31,43|b^a=b^6,c^a=c^30,d^a=d^42

Conjugacy classes & character table

4667 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₃×D₂₁₇, D₃₁×D₃₀₁, D₇×D₁₃₃₃, …
maximal normal subgroup of	D₄₃×D₂₁₇, D₃₁×D₃₀₁, D₇×D₁₃₃₃, …
maximal quotient of	Dic₉₃₃₁, C₂×D₉₃₃₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18662, 9);

D9331 #32916254