D₁₇₈₇₁ #49088357

order	35742
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₈₇₁
unambiguous semidirect product	C₃⋊D₅₉₅₇
unambiguous semidirect product	C₇⋊D₂₅₅₃
unambiguous semidirect product	C₂₃⋊D₇₇₇
unambiguous semidirect product	C₃₇⋊D₄₈₃

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₂₃×C₃₇ ⋊ C₂
Hall	C₇×C₂₃×C₃₇ ⋊ D₃
Hall	C₃×C₂₃×C₃₇ ⋊ D₇
Hall	C₂₃×C₃₇ ⋊ D₂₁
Hall	C₃×C₇×C₃₇ ⋊ D₂₃
Hall	C₃×C₇×C₂₃ ⋊ D₃₇
Hall	C₇×C₃₇ ⋊ D₆₉
Hall	C₇×C₂₃ ⋊ D₁₁₁
Hall	C₃×C₃₇ ⋊ D₁₆₁
Hall	C₃×C₂₃ ⋊ D₂₅₉
Sylow	C₃₇ ⋊ D₄₈₃
Sylow	C₂₃ ⋊ D₇₇₇
Hall	C₃×C₇ ⋊ D₈₅₁
Sylow	C₇ ⋊ D₂₅₅₃
Sylow	C₃ ⋊ D₅₉₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₂₃×C₃₇	C₂
LCS	G ▷ C₃×C₇×C₂₃×C₃₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 17871

A minimal faithful permutation representation — 2 generators on 70 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);(2,3)(5,10)(6,9)(7,8)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2,3,7,23,37|b^a=b^2,c^a=c^6,d^a=d^22,e^a=e^36

Conjugacy classes & character table

8937 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35742, 20);

D17871 #49088357