D₇₅₆₇ #29502811

order	15134
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₅₆₇
unambiguous semidirect product	C₇⋊D₁₀₈₁
unambiguous semidirect product	C₂₃⋊D₃₂₉
unambiguous semidirect product	C₄₇⋊D₁₆₁

Splittings

Fitting, socle, derived, Hall	C₇×C₂₃×C₄₇ ⋊ C₂
Hall	C₂₃×C₄₇ ⋊ D₇
Hall	C₇×C₄₇ ⋊ D₂₃
Hall	C₇×C₂₃ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₁₆₁
Sylow	C₂₃ ⋊ D₃₂₉
Sylow	C₇ ⋊ D₁₀₈₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₂₃×C₄₇	C₂
LCS	G ▷ C₇×C₂₃×C₄₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 7567

A minimal faithful permutation representation — 2 generators on 77 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);(2,7)(3,6)(4,5)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(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:

2,7,23,47|b^a=b^6,c^a=c^22,d^a=d^46

Conjugacy classes & character table

3785 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₇×D₁₆₁, D₂₃×D₃₂₉, D₇×D₁₀₈₁, …
maximal normal subgroup of	D₄₇×D₁₆₁, D₂₃×D₃₂₉, D₇×D₁₀₈₁, …
maximal quotient of	Dic₇₅₆₇, C₂×D₇₅₆₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15134, 9);

D7567 #29502811