D₁₂₂₉₉ #37589725

order	24598
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₂₉₉
unambiguous semidirect product	C₄₉⋊D₂₅₁
unambiguous semidirect product	C₂₅₁⋊D₄₉

Splittings

Fitting, derived, Hall	C₄₉×C₂₅₁ ⋊ C₂
Sylow	C₂₅₁ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₂₅₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉×C₂₅₁	C₂
Frattini	C₇	D₁₇₅₇
socle	C₇×C₂₅₁	D₇
LCS	G ▷ C₄₉×C₂₅₁
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 300
transitive degree	μ_t(G) = 12299

A minimal faithful permutation representation — 2 generators on 300 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)(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300);(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,300)(52,299)(53,298)(54,297)(55,296)(56,295)(57,294)(58,293)(59,292)(60,291)(61,290)(62,289)(63,288)(64,287)(65,286)(66,285)(67,284)(68,283)(69,282)(70,281)(71,280)(72,279)(73,278)(74,277)(75,276)(76,275)(77,274)(78,273)(79,272)(80,271)(81,270)(82,269)(83,268)(84,267)(85,266)(86,265)(87,264)(88,263)(89,262)(90,261)(91,260)(92,259)(93,258)(94,257)(95,256)(96,255)(97,254)(98,253)(99,252)(100,251)(101,250)(102,249)(103,248)(104,247)(105,246)(106,245)(107,244)(108,243)(109,242)(110,241)(111,240)(112,239)(113,238)(114,237)(115,236)(116,235)(117,234)(118,233)(119,232)(120,231)(121,230)(122,229)(123,228)(124,227)(125,226)(126,225)(127,224)(128,223)(129,222)(130,221)(131,220)(132,219)(133,218)(134,217)(135,216)(136,215)(137,214)(138,213)(139,212)(140,211)(141,210)(142,209)(143,208)(144,207)(145,206)(146,205)(147,204)(148,203)(149,202)(150,201)(151,200)(152,199)(153,198)(154,197)(155,196)(156,195)(157,194)(158,193)(159,192)(160,191)(161,190)(162,189)(163,188)(164,187)(165,186)(166,185)(167,184)(168,183)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)

Polycyclic presentation:

2,7,251,7|b^7=d^6,b^a=b^6d,c^a=c^250,d^a=d^6

Conjugacy classes & character table

6151 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24598, 4);

D12299 #37589725