D₇₆₆₁ #29506209

order	15322
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₆₆₁
unambiguous semidirect product	C₄₇⋊D₁₆₃
unambiguous semidirect product	C₁₆₃⋊D₄₇

Splittings

Fitting, socle, derived, Hall	C₄₇×C₁₆₃ ⋊ C₂
Sylow	C₁₆₃ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₁₆₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₄₇×C₁₆₃	C₂
LCS	G ▷ C₄₇×C₁₆₃
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 210
transitive degree	μ_t(G) = 7661

A minimal faithful permutation representation — 2 generators on 210 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)(48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210);(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(49,210)(50,209)(51,208)(52,207)(53,206)(54,205)(55,204)(56,203)(57,202)(58,201)(59,200)(60,199)(61,198)(62,197)(63,196)(64,195)(65,194)(66,193)(67,192)(68,191)(69,190)(70,189)(71,188)(72,187)(73,186)(74,185)(75,184)(76,183)(77,182)(78,181)(79,180)(80,179)(81,178)(82,177)(83,176)(84,175)(85,174)(86,173)(87,172)(88,171)(89,170)(90,169)(91,168)(92,167)(93,166)(94,165)(95,164)(96,163)(97,162)(98,161)(99,160)(100,159)(101,158)(102,157)(103,156)(104,155)(105,154)(106,153)(107,152)(108,151)(109,150)(110,149)(111,148)(112,147)(113,146)(114,145)(115,144)(116,143)(117,142)(118,141)(119,140)(120,139)(121,138)(122,137)(123,136)(124,135)(125,134)(126,133)(127,132)(128,131)(129,130)

Polycyclic presentation:

2,47,163|b^a=b^46,c^a=c^162

Conjugacy classes & character table

3832 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₇×D₁₆₃, C₂×D₇₆₆₁
maximal normal subgroup of	D₄₇×D₁₆₃, C₂×D₇₆₆₁
maximal quotient of	Dic₇₆₆₁, C₂×D₇₆₆₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15322, 3);

D7661 #29506209