D₁₃₁₉₉ #40995969

order	26398
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₁₉₉
unambiguous semidirect product	C₆₇⋊D₁₉₇
unambiguous semidirect product	C₁₉₇⋊D₆₇

Splittings

Fitting, socle, derived, Hall	C₆₇×C₁₉₇ ⋊ C₂
Sylow	C₁₉₇ ⋊ D₆₇
Sylow	C₆₇ ⋊ D₁₉₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₆₇×C₁₉₇	C₂
LCS	G ▷ C₆₇×C₁₉₇
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 264
transitive degree	μ_t(G) = 13199

A minimal faithful permutation representation — 2 generators on 264 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);(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)(69,264)(70,263)(71,262)(72,261)(73,260)(74,259)(75,258)(76,257)(77,256)(78,255)(79,254)(80,253)(81,252)(82,251)(83,250)(84,249)(85,248)(86,247)(87,246)(88,245)(89,244)(90,243)(91,242)(92,241)(93,240)(94,239)(95,238)(96,237)(97,236)(98,235)(99,234)(100,233)(101,232)(102,231)(103,230)(104,229)(105,228)(106,227)(107,226)(108,225)(109,224)(110,223)(111,222)(112,221)(113,220)(114,219)(115,218)(116,217)(117,216)(118,215)(119,214)(120,213)(121,212)(122,211)(123,210)(124,209)(125,208)(126,207)(127,206)(128,205)(129,204)(130,203)(131,202)(132,201)(133,200)(134,199)(135,198)(136,197)(137,196)(138,195)(139,194)(140,193)(141,192)(142,191)(143,190)(144,189)(145,188)(146,187)(147,186)(148,185)(149,184)(150,183)(151,182)(152,181)(153,180)(154,179)(155,178)(156,177)(157,176)(158,175)(159,174)(160,173)(161,172)(162,171)(163,170)(164,169)(165,168)(166,167)

Polycyclic presentation:

2,67,197|b^a=b^66,c^a=c^196

Conjugacy classes & character table

6601 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₁₉₉
maximal normal subgroup of	C₂×D₁₃₁₉₉
maximal quotient of	Dic₁₃₁₉₉, C₂×D₁₃₁₉₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26398, 3);

D13199 #40995969