D₁₁₈₀₄ #72223969

order	23608
automorphism	Aut(G) order 64024896
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₈₀₄

Splittings

Fitting	C₄×C₁₃×C₂₂₇ ⋊ C₂
Hall	C₁₃×C₂₂₇ ⋊ D₄
Sylow	C₂₂₇ ⋊ D₅₂
Sylow	C₁₃ ⋊ D₉₀₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₅₁
commutator, socle, cosocle	C₂×C₁₃×C₂₂₇	C₂²
Fitting	C₄×C₁₃×C₂₂₇	C₂
hypercenter	C₄	D₂₉₅₁
hypocenter	C₁₃×C₂₂₇	D₄
LCS	G ▷ C₂×C₁₃×C₂₂₇ ▷ C₁₃×C₂₂₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 244
transitive degree	μ_t(G) = 11804

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

Polycyclic presentation:

2^3,13,227|b^2=c,b^a=bc,d^a=d^12,e^a=e^226

Conjugacy classes & character table

5905 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₆₀₈
maximal normal subgroup of	D₂₃₆₀₈
maximal quotient of	Dic₁₁₈₀₄, D₂₃₆₀₈

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 13, 227]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^226);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11804 #72223969