Dic₂₇₀₂ #72212919

order	10808
automorphism	Aut(G) order 12450816
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₇₀₂

Splittings

Hall	C₇×C₁₉₃ ⋊ Q₈
Sylow	C₁₉₃ ⋊ Dic₁₄
Sylow	C₇ ⋊ Dic₃₈₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₃₅₁
commutator, socle, prime radical, cosocle	C₂×C₇×C₁₉₃	C₂²
Fitting	C₄×C₇×C₁₉₃	C₂
hypercenter	C₄	D₁₃₅₁
hypocenter	C₇×C₁₉₃	Q₈
LCS	G ▷ C₂×C₇×C₁₉₃ ▷ C₇×C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 208
transitive degree	μ_t(G) = 10808

A minimal faithful permutation representation — 2 generators on 208 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);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,208)(18,207)(19,206)(20,205)(21,204)(22,203)(23,202)(24,201)(25,200)(26,199)(27,198)(28,197)(29,196)(30,195)(31,194)(32,193)(33,192)(34,191)(35,190)(36,189)(37,188)(38,187)(39,186)(40,185)(41,184)(42,183)(43,182)(44,181)(45,180)(46,179)(47,178)(48,177)(49,176)(50,175)(51,174)(52,173)(53,172)(54,171)(55,170)(56,169)(57,168)(58,167)(59,166)(60,165)(61,164)(62,163)(63,162)(64,161)(65,160)(66,159)(67,158)(68,157)(69,156)(70,155)(71,154)(72,153)(73,152)(74,151)(75,150)(76,149)(77,148)(78,147)(79,146)(80,145)(81,144)(82,143)(83,142)(84,141)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,124)(102,123)(103,122)(104,121)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)

Polycyclic presentation:

2^3,7,193|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^192

Conjugacy classes & character table

2705 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₄₀₄, Dic₈₁₀₆, Dic₁₃₅₁₀
maximal normal subgroup of	Dic₅₄₀₄
maximal quotient of	Dic₈₁₀₆, Dic₁₃₅₁₀

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, 7, 193]);;
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.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^192);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2702 #72212919