Dic₂₇₄₈ #72212972

order	10992
automorphism	Aut(G) order 10024704
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₇₄₈

Splittings

Hall	C₃×C₂₂₉ ⋊ Q₁₆
Sylow	C₂₂₉ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₉₁₆

Subgroups

	subgroup	G/H
center	C₂	D₂₇₄₈
commutator, cosocle	C₃×C₄×C₂₂₉	C₂²
Frattini	C₄	C₂×D₆₈₇
Fitting	C₃×C₈×C₂₂₉	C₂
socle, prime radical	C₂×C₃×C₂₂₉	D₄
hypercenter	C₈	D₆₈₇
hypocenter	C₃×C₂₂₉	Q₁₆
LCS	G ▷ C₃×C₄×C₂₂₉ ▷ C₂×C₃×C₂₂₉ ▷ C₃×C₂₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 248
transitive degree	μ_t(G) = 10992

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

Polycyclic presentation:

2^4,3,229|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^228

Conjugacy classes & character table

2751 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₄₉₆, Dic₈₂₄₄, Dic₁₃₇₄₀
maximal normal subgroup of	Dic₅₄₉₆
maximal quotient of	Dic₈₂₄₄, Dic₁₃₇₄₀

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, 3, 229]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^228);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2748 #72212972