D₅₄₆₇ #24926451

order	10934
automorphism	Aut(G) order 22961400
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₅₄₆₇
unambiguous semidirect product	C₇⋊D₇₈₁
unambiguous semidirect product	C₁₁⋊D₄₉₇
unambiguous semidirect product	C₇₁⋊D₇₇

Splittings

Fitting, socle, derived, Hall	C₇×C₁₁×C₇₁ ⋊ C₂
Hall	C₁₁×C₇₁ ⋊ D₇
Hall	C₇×C₇₁ ⋊ D₁₁
Hall	C₇×C₁₁ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₇₇
Sylow	C₁₁ ⋊ D₄₉₇
Sylow	C₇ ⋊ D₇₈₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₁×C₇₁	C₂
LCS	G ▷ C₇×C₁₁×C₇₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 5467

A minimal faithful permutation representation — 2 generators on 89 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);(2,7)(3,6)(4,5)(9,18)(10,17)(11,16)(12,15)(13,14)(20,89)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2,7,11,71|b^a=b^6,c^a=c^10,d^a=d^70

Conjugacy classes & character table

2735 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₁×D₇₇, D₁₁×D₄₉₇, D₇×D₇₈₁, …
maximal normal subgroup of	D₇₁×D₇₇, D₁₁×D₄₉₇, D₇×D₇₈₁, …
maximal quotient of	Dic₅₄₆₇, C₂×D₅₄₆₇, C₃×D₅₄₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10934, 9);

D5467 #24926451