D₁₅₃₉₇ #44502548

order	30794
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₃₉₇
unambiguous semidirect product	C₈₉⋊D₁₇₃
unambiguous semidirect product	C₁₇₃⋊D₈₉

Splittings

Fitting, socle, derived, Hall	C₈₉×C₁₇₃ ⋊ C₂
Sylow	C₁₇₃ ⋊ D₈₉
Sylow	C₈₉ ⋊ D₁₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₈₉×C₁₇₃	C₂
LCS	G ▷ C₈₉×C₁₇₃
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 262
transitive degree	μ_t(G) = 15397

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

Polycyclic presentation:

2,89,173|b^a=b^88,c^a=c^172

Conjugacy classes & character table

7700 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₃₉₇
maximal normal subgroup of	C₂×D₁₅₃₉₇
maximal quotient of	Dic₁₅₃₉₇, C₂×D₁₅₃₉₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30794, 3);

D15397 #44502548