C₂×D₆₉₃₉ #72235604

order	27756
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₃₈₇₈
direct product	C₂×D₆₉₃₉

Splittings

Fitting	C₂×C₂₇×C₂₅₇ ⋊ C₂
derived, Hall	C₂₇×C₂₅₇ ⋊ C₂²
Sylow	C₂₅₇ ⋊ C₂×D₂₇
Sylow	C₂₇ ⋊ C₂×D₂₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₉₃₉
commutator, cosocle, hypocenter	C₂₇×C₂₅₇	C₂²
Frattini	C₉	C₂×D₇₇₁
Fitting	C₂×C₂₇×C₂₅₇	C₂
socle	C₂×C₃×C₂₅₇	D₉
LCS	G ▷ C₂₇×C₂₅₇
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 286
transitive degree	μ_t(G) = 13878

A minimal faithful permutation representation — 2 generators on 286 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);(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,286)(32,285)(33,284)(34,283)(35,282)(36,281)(37,280)(38,279)(39,278)(40,277)(41,276)(42,275)(43,274)(44,273)(45,272)(46,271)(47,270)(48,269)(49,268)(50,267)(51,266)(52,265)(53,264)(54,263)(55,262)(56,261)(57,260)(58,259)(59,258)(60,257)(61,256)(62,255)(63,254)(64,253)(65,252)(66,251)(67,250)(68,249)(69,248)(70,247)(71,246)(72,245)(73,244)(74,243)(75,242)(76,241)(77,240)(78,239)(79,238)(80,237)(81,236)(82,235)(83,234)(84,233)(85,232)(86,231)(87,230)(88,229)(89,228)(90,227)(91,226)(92,225)(93,224)(94,223)(95,222)(96,221)(97,220)(98,219)(99,218)(100,217)(101,216)(102,215)(103,214)(104,213)(105,212)(106,211)(107,210)(108,209)(109,208)(110,207)(111,206)(112,205)(113,204)(114,203)(115,202)(116,201)(117,200)(118,199)(119,198)(120,197)(121,196)(122,195)(123,194)(124,193)(125,192)(126,191)(127,190)(128,189)(129,188)(130,187)(131,186)(132,185)(133,184)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,171)(147,170)(148,169)(149,168)(150,167)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(157,160)(158,159)

Conjugacy classes & character table

6942 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₇₅₆
maximal normal subgroup of	D₂₇₇₅₆
maximal quotient of	Dic₁₃₈₇₈, D₂₇₇₅₆

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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286), (4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,286)(32,285)(33,284)(34,283)(35,282)(36,281)(37,280)(38,279)(39,278)(40,277)(41,276)(42,275)(43,274)(44,273)(45,272)(46,271)(47,270)(48,269)(49,268)(50,267)(51,266)(52,265)(53,264)(54,263)(55,262)(56,261)(57,260)(58,259)(59,258)(60,257)(61,256)(62,255)(63,254)(64,253)(65,252)(66,251)(67,250)(68,249)(69,248)(70,247)(71,246)(72,245)(73,244)(74,243)(75,242)(76,241)(77,240)(78,239)(79,238)(80,237)(81,236)(82,235)(83,234)(84,233)(85,232)(86,231)(87,230)(88,229)(89,228)(90,227)(91,226)(92,225)(93,224)(94,223)(95,222)(96,221)(97,220)(98,219)(99,218)(100,217)(101,216)(102,215)(103,214)(104,213)(105,212)(106,211)(107,210)(108,209)(109,208)(110,207)(111,206)(112,205)(113,204)(114,203)(115,202)(116,201)(117,200)(118,199)(119,198)(120,197)(121,196)(122,195)(123,194)(124,193)(125,192)(126,191)(127,190)(128,189)(129,188)(130,187)(131,186)(132,185)(133,184)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,171)(147,170)(148,169)(149,168)(150,167)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(157,160)(158,159));

C2×D6939 #72235604