Dic₄₃₃₄ #72215769

order	17336
automorphism	Aut(G) order 33978560
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₃₄

Splittings

Hall	C₁₁×C₁₉₇ ⋊ Q₈
Sylow	C₁₉₇ ⋊ Dic₂₂
Sylow	C₁₁ ⋊ Dic₃₉₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₆₇
commutator, socle, prime radical, cosocle	C₂×C₁₁×C₁₉₇	C₂²
Fitting	C₄×C₁₁×C₁₉₇	C₂
hypercenter	C₄	D₂₁₆₇
hypocenter	C₁₁×C₁₉₇	Q₈
LCS	G ▷ C₂×C₁₁×C₁₉₇ ▷ C₁₁×C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 216
transitive degree	μ_t(G) = 17336

A minimal faithful permutation representation — 2 generators on 216 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,19)(11,18)(12,17)(13,16)(14,15)(21,216)(22,215)(23,214)(24,213)(25,212)(26,211)(27,210)(28,209)(29,208)(30,207)(31,206)(32,205)(33,204)(34,203)(35,202)(36,201)(37,200)(38,199)(39,198)(40,197)(41,196)(42,195)(43,194)(44,193)(45,192)(46,191)(47,190)(48,189)(49,188)(50,187)(51,186)(52,185)(53,184)(54,183)(55,182)(56,181)(57,180)(58,179)(59,178)(60,177)(61,176)(62,175)(63,174)(64,173)(65,172)(66,171)(67,170)(68,169)(69,168)(70,167)(71,166)(72,165)(73,164)(74,163)(75,162)(76,161)(77,160)(78,159)(79,158)(80,157)(81,156)(82,155)(83,154)(84,153)(85,152)(86,151)(87,150)(88,149)(89,148)(90,147)(91,146)(92,145)(93,144)(94,143)(95,142)(96,141)(97,140)(98,139)(99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)

Polycyclic presentation:

2^3,11,197|a^2=c,b^2=c,b^a=bc,d^a=d^10,e^a=e^196

Conjugacy classes & character table

4337 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₆₆₈, Dic₁₃₀₀₂
maximal normal subgroup of	Dic₈₆₆₈
maximal quotient of	Dic₁₃₀₀₂

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, 11, 197]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^196);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4334 #72215769