Dic₂₃₈₈ #72212717

order	9552
automorphism	Aut(G) order 7565184
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₃₈₈

Splittings

Hall	C₃×C₁₉₉ ⋊ Q₁₆
Sylow	C₁₉₉ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₇₉₆

Subgroups

	subgroup	G/H
center	C₂	D₂₃₈₈
commutator, cosocle	C₃×C₄×C₁₉₉	C₂²
Frattini	C₄	C₂×D₅₉₇
Fitting	C₃×C₈×C₁₉₉	C₂
socle, prime radical	C₂×C₃×C₁₉₉	D₄
hypercenter	C₈	D₅₉₇
hypocenter	C₃×C₁₉₉	Q₁₆
LCS	G ▷ C₃×C₄×C₁₉₉ ▷ C₂×C₃×C₁₉₉ ▷ C₃×C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 218
transitive degree	μ_t(G) = 9552

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

Polycyclic presentation:

2^4,3,199|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^198

Conjugacy classes & character table

2391 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₇₇₆, Dic₇₁₆₄, Dic₁₁₉₄₀
maximal normal subgroup of	Dic₄₇₇₆
maximal quotient of	Dic₇₁₆₄, Dic₁₁₉₄₀

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, 199]);;
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.6^198);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2388 #72212717