C₂×D₁₆₁₉₉ #72302115

order	64796
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₂₃₉₈
direct product	C₂×D₁₆₁₉₉

Splittings

Fitting, socle	C₂×C₉₇×C₁₆₇ ⋊ C₂
derived, Hall	C₉₇×C₁₆₇ ⋊ C₂²
Sylow	C₁₆₇ ⋊ C₂×D₉₇
Sylow	C₉₇ ⋊ C₂×D₁₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₆₁₉₉
commutator, cosocle, hypocenter	C₉₇×C₁₆₇	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₉₇×C₁₆₇	C₂
LCS	G ▷ C₉₇×C₁₆₇
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 266
transitive degree	μ_t(G) = 32398

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

Conjugacy classes & character table

16202 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C2×D16199 #72302115