D₁₅₂₀₉ #44477409

order	30418
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₂₀₉
unambiguous semidirect product	C₆₇⋊D₂₂₇
unambiguous semidirect product	C₂₂₇⋊D₆₇

Splittings

Fitting, socle, derived, Hall	C₆₇×C₂₂₇ ⋊ C₂
Sylow	C₂₂₇ ⋊ D₆₇
Sylow	C₆₇ ⋊ D₂₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₆₇×C₂₂₇	C₂
LCS	G ▷ C₆₇×C₂₂₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 294
transitive degree	μ_t(G) = 15209

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

Polycyclic presentation:

2,67,227|b^a=b^66,c^a=c^226

Conjugacy classes & character table

7606 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₂₀₉
maximal normal subgroup of	C₂×D₁₅₂₀₉
maximal quotient of	Dic₁₅₂₀₉, C₂×D₁₅₂₀₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30418, 3);

D15209 #44477409