D₁₇₁₅₇ #46804902

order	34314
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₁₅₇
unambiguous semidirect product	C₃⋊D₅₇₁₉
unambiguous semidirect product	C₇⋊D₂₄₅₁
unambiguous semidirect product	C₁₉⋊D₉₀₃
unambiguous semidirect product	C₄₃⋊D₃₉₉

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₁₉×C₄₃ ⋊ C₂
Hall	C₇×C₁₉×C₄₃ ⋊ D₃
Hall	C₃×C₁₉×C₄₃ ⋊ D₇
Hall	C₃×C₇×C₄₃ ⋊ D₁₉
Hall	C₁₉×C₄₃ ⋊ D₂₁
Hall	C₃×C₇×C₁₉ ⋊ D₄₃
Hall	C₇×C₄₃ ⋊ D₅₇
Hall	C₇×C₁₉ ⋊ D₁₂₉
Hall	C₃×C₄₃ ⋊ D₁₃₃
Hall	C₃×C₁₉ ⋊ D₃₀₁
Sylow	C₄₃ ⋊ D₃₉₉
Hall	C₃×C₇ ⋊ D₈₁₇
Sylow	C₁₉ ⋊ D₉₀₃
Sylow	C₇ ⋊ D₂₄₅₁
Sylow	C₃ ⋊ D₅₇₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₉×C₄₃	C₂
LCS	G ▷ C₃×C₇×C₁₉×C₄₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 17157

A minimal faithful permutation representation — 2 generators on 72 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);(2,3)(5,10)(6,9)(7,8)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2,3,7,19,43|b^a=b^2,c^a=c^6,d^a=d^18,e^a=e^42

Conjugacy classes & character table

8580 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34314, 34);

D17157 #46804902