Dic₇₅₈₈ #72271970

order	30352
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₅₈₈

Splittings

Hall	C₇×C₂₇₁ ⋊ Q₁₆
Sylow	C₂₇₁ ⋊ Dic₂₈
Sylow	C₇ ⋊ Dic₁₀₈₄

Subgroups

	subgroup	G/H
center	C₂	D₇₅₈₈
commutator, cosocle	C₄×C₇×C₂₇₁	C₂²
Frattini	C₄	C₂×D₁₈₉₇
Fitting	C₇×C₈×C₂₇₁	C₂
socle, prime radical	C₂×C₇×C₂₇₁	D₄
hypercenter	C₈	D₁₈₉₇
hypocenter	C₇×C₂₇₁	Q₁₆
LCS	G ▷ C₄×C₇×C₂₇₁ ▷ C₂×C₇×C₂₇₁ ▷ C₇×C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 294
transitive degree	μ_t(G) = 30352

A minimal faithful permutation representation — 2 generators on 294 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,23)(19,22)(20,21)(25,294)(26,293)(27,292)(28,291)(29,290)(30,289)(31,288)(32,287)(33,286)(34,285)(35,284)(36,283)(37,282)(38,281)(39,280)(40,279)(41,278)(42,277)(43,276)(44,275)(45,274)(46,273)(47,272)(48,271)(49,270)(50,269)(51,268)(52,267)(53,266)(54,265)(55,264)(56,263)(57,262)(58,261)(59,260)(60,259)(61,258)(62,257)(63,256)(64,255)(65,254)(66,253)(67,252)(68,251)(69,250)(70,249)(71,248)(72,247)(73,246)(74,245)(75,244)(76,243)(77,242)(78,241)(79,240)(80,239)(81,238)(82,237)(83,236)(84,235)(85,234)(86,233)(87,232)(88,231)(89,230)(90,229)(91,228)(92,227)(93,226)(94,225)(95,224)(96,223)(97,222)(98,221)(99,220)(100,219)(101,218)(102,217)(103,216)(104,215)(105,214)(106,213)(107,212)(108,211)(109,210)(110,209)(111,208)(112,207)(113,206)(114,205)(115,204)(116,203)(117,202)(118,201)(119,200)(120,199)(121,198)(122,197)(123,196)(124,195)(125,194)(126,193)(127,192)(128,191)(129,190)(130,189)(131,188)(132,187)(133,186)(134,185)(135,184)(136,183)(137,182)(138,181)(139,180)(140,179)(141,178)(142,177)(143,176)(144,175)(145,174)(146,173)(147,172)(148,171)(149,170)(150,169)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)

Polycyclic presentation:

2^4,7,271|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^270

Conjugacy classes & character table

7591 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₁₇₆
maximal normal subgroup of	Dic₁₅₁₇₆

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, 7, 271]);;
gap> SetPower(coll, 1, F.4);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^270);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7588 #72271970