D₁₃₂₁₃ #40996173

order	26426
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₂₁₃
unambiguous semidirect product	C₇₃⋊D₁₈₁
unambiguous semidirect product	C₁₈₁⋊D₇₃

Splittings

Fitting, socle, derived, Hall	C₇₃×C₁₈₁ ⋊ C₂
Sylow	C₁₈₁ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₁₈₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇₃×C₁₈₁	C₂
LCS	G ▷ C₇₃×C₁₈₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 254
transitive degree	μ_t(G) = 13213

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

Polycyclic presentation:

2,73,181|b^a=b^72,c^a=c^180

Conjugacy classes & character table

6608 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₂₁₃
maximal normal subgroup of	C₂×D₁₃₂₁₃
maximal quotient of	Dic₁₃₂₁₃, C₂×D₁₃₂₁₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26426, 3);

D13213 #40996173