D₁₈₉₉₇ #49184353

order	37994
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₈₉₉₇
unambiguous semidirect product	C₁₂₁⋊D₁₅₇
unambiguous semidirect product	C₁₅₇⋊D₁₂₁

Splittings

Fitting, derived, Hall	C₁₂₁×C₁₅₇ ⋊ C₂
Sylow	C₁₅₇ ⋊ D₁₂₁
Sylow	C₁₂₁ ⋊ D₁₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₂₁×C₁₅₇	C₂
Frattini	C₁₁	D₁₇₂₇
socle	C₁₁×C₁₅₇	D₁₁
LCS	G ▷ C₁₂₁×C₁₅₇
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 278
transitive degree	μ_t(G) = 18997

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

Polycyclic presentation:

2,11,157,11|b^11=d^10,b^a=b^10d,c^a=c^156,d^a=d^10

Conjugacy classes & character table

9500 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37994, 4);

D18997 #49184353