C₂×D₅₆₉₇ #72215944

order	22788
automorphism	Aut(G) order 43069320
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₁₃₉₄
direct product	C₂×D₅₆₉₇

Splittings

Fitting	C₂×C₂₇×C₂₁₁ ⋊ C₂
derived, Hall	C₂₇×C₂₁₁ ⋊ C₂²
Sylow	C₂₁₁ ⋊ C₂×D₂₇
Sylow	C₂₇ ⋊ C₂×D₂₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₆₉₇
commutator, cosocle, hypocenter	C₂₇×C₂₁₁	C₂²
Frattini	C₉	C₂×D₆₃₃
Fitting	C₂×C₂₇×C₂₁₁	C₂
socle	C₂×C₃×C₂₁₁	D₉
LCS	G ▷ C₂₇×C₂₁₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 240
transitive degree	μ_t(G) = 11394

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

Conjugacy classes & character table

5700 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₇₈₈
maximal normal subgroup of	D₂₂₇₈₈
maximal quotient of	Dic₁₁₃₉₄, D₂₂₇₈₈

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), (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)(31,240)(32,239)(33,238)(34,237)(35,236)(36,235)(37,234)(38,233)(39,232)(40,231)(41,230)(42,229)(43,228)(44,227)(45,226)(46,225)(47,224)(48,223)(49,222)(50,221)(51,220)(52,219)(53,218)(54,217)(55,216)(56,215)(57,214)(58,213)(59,212)(60,211)(61,210)(62,209)(63,208)(64,207)(65,206)(66,205)(67,204)(68,203)(69,202)(70,201)(71,200)(72,199)(73,198)(74,197)(75,196)(76,195)(77,194)(78,193)(79,192)(80,191)(81,190)(82,189)(83,188)(84,187)(85,186)(86,185)(87,184)(88,183)(89,182)(90,181)(91,180)(92,179)(93,178)(94,177)(95,176)(96,175)(97,174)(98,173)(99,172)(100,171)(101,170)(102,169)(103,168)(104,167)(105,166)(106,165)(107,164)(108,163)(109,162)(110,161)(111,160)(112,159)(113,158)(114,157)(115,156)(116,155)(117,154)(118,153)(119,152)(120,151)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,144)(128,143)(129,142)(130,141)(131,140)(132,139)(133,138)(134,137)(135,136));

C2×D5697 #72215944