D₁₁₆₀₇ #37551003

order	23214
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₆₀₇
unambiguous semidirect product	C₃⋊D₃₈₆₉
unambiguous semidirect product	C₅₃⋊D₂₁₉
unambiguous semidirect product	C₇₃⋊D₁₅₉

Splittings

Fitting, socle, derived, Hall	C₃×C₅₃×C₇₃ ⋊ C₂
Hall	C₅₃×C₇₃ ⋊ D₃
Hall	C₃×C₇₃ ⋊ D₅₃
Hall	C₃×C₅₃ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₁₅₉
Sylow	C₅₃ ⋊ D₂₁₉
Sylow	C₃ ⋊ D₃₈₆₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅₃×C₇₃	C₂
LCS	G ▷ C₃×C₅₃×C₇₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 11607

A minimal faithful permutation representation — 2 generators on 129 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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(2,3)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(73,114)(74,113)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)

Polycyclic presentation:

2,3,53,73|b^a=b^2,c^a=c^52,d^a=d^72

Conjugacy classes & character table

5805 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23214, 9);

D11607 #37551003