Dic₁₃₉₃₀ #72290267

order	55720
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₉₃₀

Splittings

Hall	C₅×C₇×C₁₉₉ ⋊ Q₈
Hall	C₇×C₁₉₉ ⋊ Dic₁₀
Hall	C₅×C₁₉₉ ⋊ Dic₁₄
Sylow	C₁₉₉ ⋊ Dic₇₀
Hall	C₅×C₇ ⋊ Dic₃₉₈
Sylow	C₇ ⋊ Dic₁₉₉₀
Sylow	C₅ ⋊ Dic₂₇₈₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₉₆₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₇×C₁₉₉	C₂²
Fitting	C₄×C₅×C₇×C₁₉₉	C₂
hypercenter	C₄	D₆₉₆₅
hypocenter	C₅×C₇×C₁₉₉	Q₈
LCS	G ▷ C₂×C₅×C₇×C₁₉₉ ▷ C₅×C₇×C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 219
transitive degree	μ_t(G) = 55720

A minimal faithful permutation representation — 2 generators on 219 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13)(14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,20)(16,19)(17,18)(22,219)(23,218)(24,217)(25,216)(26,215)(27,214)(28,213)(29,212)(30,211)(31,210)(32,209)(33,208)(34,207)(35,206)(36,205)(37,204)(38,203)(39,202)(40,201)(41,200)(42,199)(43,198)(44,197)(45,196)(46,195)(47,194)(48,193)(49,192)(50,191)(51,190)(52,189)(53,188)(54,187)(55,186)(56,185)(57,184)(58,183)(59,182)(60,181)(61,180)(62,179)(63,178)(64,177)(65,176)(66,175)(67,174)(68,173)(69,172)(70,171)(71,170)(72,169)(73,168)(74,167)(75,166)(76,165)(77,164)(78,163)(79,162)(80,161)(81,160)(82,159)(83,158)(84,157)(85,156)(86,155)(87,154)(88,153)(89,152)(90,151)(91,150)(92,149)(93,148)(94,147)(95,146)(96,145)(97,144)(98,143)(99,142)(100,141)(101,140)(102,139)(103,138)(104,137)(105,136)(106,135)(107,134)(108,133)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121)

Polycyclic presentation:

2^3,5,7,199|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^198

Conjugacy classes & character table

13933 conjugacy classes. The character table is too large to display.

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, 5, 7, 199]);;
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.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^198);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13930 #72290267