D₁₅₇₂₁ #44537735

order	31442
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₇₂₁
unambiguous semidirect product	C₇₉⋊D₁₉₉
unambiguous semidirect product	C₁₉₉⋊D₇₉

Splittings

Fitting, socle, derived, Hall	C₇₉×C₁₉₉ ⋊ C₂
Sylow	C₁₉₉ ⋊ D₇₉
Sylow	C₇₉ ⋊ D₁₉₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇₉×C₁₉₉	C₂
LCS	G ▷ C₇₉×C₁₉₉
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 278
transitive degree	μ_t(G) = 15721

A minimal faithful permutation representation — 2 generators on 278 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79)(80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278);(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(81,278)(82,277)(83,276)(84,275)(85,274)(86,273)(87,272)(88,271)(89,270)(90,269)(91,268)(92,267)(93,266)(94,265)(95,264)(96,263)(97,262)(98,261)(99,260)(100,259)(101,258)(102,257)(103,256)(104,255)(105,254)(106,253)(107,252)(108,251)(109,250)(110,249)(111,248)(112,247)(113,246)(114,245)(115,244)(116,243)(117,242)(118,241)(119,240)(120,239)(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,232)(128,231)(129,230)(130,229)(131,228)(132,227)(133,226)(134,225)(135,224)(136,223)(137,222)(138,221)(139,220)(140,219)(141,218)(142,217)(143,216)(144,215)(145,214)(146,213)(147,212)(148,211)(149,210)(150,209)(151,208)(152,207)(153,206)(154,205)(155,204)(156,203)(157,202)(158,201)(159,200)(160,199)(161,198)(162,197)(163,196)(164,195)(165,194)(166,193)(167,192)(168,191)(169,190)(170,189)(171,188)(172,187)(173,186)(174,185)(175,184)(176,183)(177,182)(178,181)(179,180)

Polycyclic presentation:

2,79,199|b^a=b^78,c^a=c^198

Conjugacy classes & character table

7862 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₇₂₁
maximal normal subgroup of	C₂×D₁₅₇₂₁
maximal quotient of	Dic₁₅₇₂₁, C₂×D₁₅₇₂₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31442, 3);

D15721 #44537735