D₃₀₈₇₂ #72297075

order	61744
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₈₇₂

Splittings

Fitting	C₈×C₁₇×C₂₂₇ ⋊ C₂
Hall	C₁₇×C₂₂₇ ⋊ D₈
Sylow	C₂₂₇ ⋊ D₁₃₆
Sylow	C₁₇ ⋊ D₁₈₁₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₄₃₆
commutator, cosocle	C₄×C₁₇×C₂₂₇	C₂²
Frattini	C₄	C₂×D₃₈₅₉
Fitting	C₈×C₁₇×C₂₂₇	C₂
socle	C₂×C₁₇×C₂₂₇	D₄
hypercenter	C₈	D₃₈₅₉
hypocenter	C₁₇×C₂₂₇	D₈
LCS	G ▷ C₄×C₁₇×C₂₂₇ ▷ C₂×C₁₇×C₂₂₇ ▷ C₁₇×C₂₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 252
transitive degree	μ_t(G) = 30872

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

Polycyclic presentation:

2^4,17,227|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^16,f^a=f^226

Conjugacy classes & character table

15439 conjugacy classes. The character table is too large to display.

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 17, 227]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^226);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30872 #72297075