C₂×D₅₅₈₉ #72215890

order	22356
automorphism	Aut(G) order 39838392
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₁₁₇₈
direct product	C₂×D₅₅₈₉

Splittings

Fitting	C₂×C₂₃×C₂₄₃ ⋊ C₂
derived, Hall	C₂₃×C₂₄₃ ⋊ C₂²
Sylow	C₂₄₃ ⋊ C₂×D₂₃
Sylow	C₂₃ ⋊ C₂×D₂₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₅₈₉
commutator, cosocle, hypocenter	C₂₃×C₂₄₃	C₂²
Frattini	C₈₁	C₂×D₆₉
Fitting	C₂×C₂₃×C₂₄₃	C₂
socle	C₂×C₃×C₂₃	D₈₁
LCS	G ▷ C₂₃×C₂₄₃
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 268
transitive degree	μ_t(G) = 11178

A minimal faithful permutation representation — 2 generators on 268 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);(4,245)(5,244)(6,243)(7,242)(8,241)(9,240)(10,239)(11,238)(12,237)(13,236)(14,235)(15,234)(16,233)(17,232)(18,231)(19,230)(20,229)(21,228)(22,227)(23,226)(24,225)(25,224)(26,223)(27,222)(28,221)(29,220)(30,219)(31,218)(32,217)(33,216)(34,215)(35,214)(36,213)(37,212)(38,211)(39,210)(40,209)(41,208)(42,207)(43,206)(44,205)(45,204)(46,203)(47,202)(48,201)(49,200)(50,199)(51,198)(52,197)(53,196)(54,195)(55,194)(56,193)(57,192)(58,191)(59,190)(60,189)(61,188)(62,187)(63,186)(64,185)(65,184)(66,183)(67,182)(68,181)(69,180)(70,179)(71,178)(72,177)(73,176)(74,175)(75,174)(76,173)(77,172)(78,171)(79,170)(80,169)(81,168)(82,167)(83,166)(84,165)(85,164)(86,163)(87,162)(88,161)(89,160)(90,159)(91,158)(92,157)(93,156)(94,155)(95,154)(96,153)(97,152)(98,151)(99,150)(100,149)(101,148)(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)(109,140)(110,139)(111,138)(112,137)(113,136)(114,135)(115,134)(116,133)(117,132)(118,131)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(247,268)(248,267)(249,266)(250,265)(251,264)(252,263)(253,262)(254,261)(255,260)(256,259)(257,258)

Conjugacy classes & character table

5592 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₃₅₆
maximal normal subgroup of	D₂₂₃₅₆
maximal quotient of	Dic₁₁₁₇₈, D₂₂₃₅₆

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

C2×D5589 #72215890