D₁₁₁₈₄ #72215870

order	22368
automorphism	Aut(G) order 41515008
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₈₄

Splittings

Fitting	C₃×C₁₆×C₂₃₃ ⋊ C₂
Hall	C₃×C₂₃₃ ⋊ D₁₆
Sylow	C₂₃₃ ⋊ D₄₈
Sylow	C₃ ⋊ D₃₇₂₈

Subgroups

	subgroup	G/H
center	C₂	D₅₅₉₂
commutator, cosocle	C₃×C₈×C₂₃₃	C₂²
Frattini	C₈	C₂×D₆₉₉
Fitting	C₃×C₁₆×C₂₃₃	C₂
socle	C₂×C₃×C₂₃₃	D₈
hypercenter	C₁₆	D₆₉₉
hypocenter	C₃×C₂₃₃	D₁₆
LCS	G ▷ C₃×C₈×C₂₃₃ ▷ C₃×C₄×C₂₃₃ ▷ C₂×C₃×C₂₃₃ ▷ C₃×C₂₃₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 252
transitive degree	μ_t(G) = 11184

A minimal faithful permutation representation — 2 generators on 252 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,252)(22,251)(23,250)(24,249)(25,248)(26,247)(27,246)(28,245)(29,244)(30,243)(31,242)(32,241)(33,240)(34,239)(35,238)(36,237)(37,236)(38,235)(39,234)(40,233)(41,232)(42,231)(43,230)(44,229)(45,228)(46,227)(47,226)(48,225)(49,224)(50,223)(51,222)(52,221)(53,220)(54,219)(55,218)(56,217)(57,216)(58,215)(59,214)(60,213)(61,212)(62,211)(63,210)(64,209)(65,208)(66,207)(67,206)(68,205)(69,204)(70,203)(71,202)(72,201)(73,200)(74,199)(75,198)(76,197)(77,196)(78,195)(79,194)(80,193)(81,192)(82,191)(83,190)(84,189)(85,188)(86,187)(87,186)(88,185)(89,184)(90,183)(91,182)(92,181)(93,180)(94,179)(95,178)(96,177)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,168)(106,167)(107,166)(108,165)(109,164)(110,163)(111,162)(112,161)(113,160)(114,159)(115,158)(116,157)(117,156)(118,155)(119,154)(120,153)(121,152)(122,151)(123,150)(124,149)(125,148)(126,147)(127,146)(128,145)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)

Polycyclic presentation:

2^5,3,233|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^232

Conjugacy classes & character table

5595 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₃₆₈
maximal normal subgroup of	D₂₂₃₆₈
maximal quotient of	Dic₁₁₁₈₄, D₂₂₃₆₈

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 233]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^232);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11184 #72215870