#27212431

order	13472
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	Q₈×C₄₂₁ ⋊ C₄
Sylow	C₄₂₁ ⋊ #104

Subgroups

	subgroup	G/H
center	C₂	C₂²⋊#13132997
commutator	C₄×C₄₂₁	C₂×C₄
Frattini	C₄	C₂×#13132997
Fitting	Q₈×C₄₂₁	C₄
socle	C₂×C₄₂₁	C₂²⋊C₄
prime radical	#20269772	C₂
cosocle	C₄×D₄₂₁	C₂²
hypercenter	Q₈	#13132997
hypocenter	C₄₂₁	#104
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 429
transitive degree	μ_t(G) = 3368

A minimal faithful permutation representation — 2 generators on 429 points:

(1,2)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,41)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,62)(31,32)(34,58)(35,36)(38,54)(39,40)(42,50)(43,44)(45,46)(47,87)(48,49)(51,83)(52,53)(55,79)(56,57)(59,75)(60,61)(63,71)(64,65)(66,67)(68,116)(69,70)(72,112)(73,74)(76,108)(77,78)(80,104)(81,82)(84,100)(85,86)(88,96)(89,90)(91,92)(93,149)(94,95)(97,145)(98,99)(101,141)(102,103)(105,137)(106,107)(109,133)(110,111)(113,129)(114,115)(117,125)(118,119)(120,121)(122,186)(123,124)(126,182)(127,128)(130,178)(131,132)(134,174)(135,136)(138,170)(139,140)(142,166)(143,144)(146,162)(147,148)(150,158)(151,152)(153,154)(155,227)(156,157)(159,223)(160,161)(163,219)(164,165)(167,215)(168,169)(171,211)(172,173)(175,207)(176,177)(179,203)(180,181)(183,199)(184,185)(187,195)(188,189)(190,191)(192,272)(193,194)(196,268)(197,198)(200,264)(201,202)(204,260)(205,206)(208,256)(209,210)(212,252)(213,214)(216,248)(217,218)(220,244)(221,222)(224,240)(225,226)(228,236)(229,230)(231,232)(233,321)(234,235)(237,317)(238,239)(241,313)(242,243)(245,309)(246,247)(249,305)(250,251)(253,301)(254,255)(257,297)(258,259)(261,293)(262,263)(265,289)(266,267)(269,285)(270,271)(273,281)(274,275)(276,277)(278,374)(279,280)(282,370)(283,284)(286,366)(287,288)(290,362)(291,292)(294,358)(295,296)(298,354)(299,300)(302,350)(303,304)(306,346)(307,308)(310,342)(311,312)(314,338)(315,316)(318,334)(319,320)(322,330)(323,324)(325,326)(327,420)(328,329)(331,418)(332,333)(335,416)(336,337)(339,414)(340,341)(343,412)(344,345)(347,410)(348,349)(351,407)(352,353)(355,403)(356,357)(359,399)(360,361)(363,395)(364,365)(367,391)(368,369)(371,387)(372,373)(375,383)(376,377)(378,379)(380,421)(381,382)(384,419)(385,386)(388,417)(389,390)(392,415)(393,394)(396,413)(397,398)(400,411)(401,402)(404,408)(405,406)(422,424)(423,426)(425,427)(428,429);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,91)(53,72,73,87)(57,76,77,83)(61,80,81,79)(65,84,85,75)(67,88,89,71)(70,93,94,120)(74,97,98,116)(78,101,102,112)(82,105,106,108)(86,109,110,104)(90,113,114,100)(92,117,118,96)(95,122,123,153)(99,126,127,149)(103,130,131,145)(107,134,135,141)(111,138,139,137)(115,142,143,133)(119,146,147,129)(121,150,151,125)(124,155,156,190)(128,159,160,186)(132,163,164,182)(136,167,168,178)(140,171,172,174)(144,175,176,170)(148,179,180,166)(152,183,184,162)(154,187,188,158)(157,192,193,231)(161,196,197,227)(165,200,201,223)(169,204,205,219)(173,208,209,215)(177,212,213,211)(181,216,217,207)(185,220,221,203)(189,224,225,199)(191,228,229,195)(194,233,234,276)(198,237,238,272)(202,241,242,268)(206,245,246,264)(210,249,250,260)(214,253,254,256)(218,257,258,252)(222,261,262,248)(226,265,266,244)(230,269,270,240)(232,273,274,236)(235,278,279,325)(239,282,283,321)(243,286,287,317)(247,290,291,313)(251,294,295,309)(255,298,299,305)(259,302,303,301)(263,306,307,297)(267,310,311,293)(271,314,315,289)(275,318,319,285)(277,322,323,281)(280,327,328,378)(284,331,332,374)(288,335,336,370)(292,339,340,366)(296,343,344,362)(300,347,348,358)(304,351,352,354)(308,355,356,350)(312,359,360,346)(316,363,364,342)(320,367,368,338)(324,371,372,334)(326,375,376,330)(329,380,381,382)(333,384,385,420)(337,388,389,418)(341,392,393,416)(345,396,397,414)(349,400,401,412)(353,404,405,410)(357,408,409,407)(361,411,406,403)(365,413,402,399)(369,415,398,395)(373,417,394,391)(377,419,390,387)(379,421,386,383)(422,423,425,428,426,424,429,427)

Polycyclic presentation:

2^5,421|a^2=d,c^2=e,d^2=e,b^a=bc,c^a=ce,f^a=f^29,c^b=ce,f^b=f^420,f^d=f^420

Conjugacy classes & character table

539 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄₂₁⋊#104, C₂×#27212431, #41003837, …
maximal normal subgroup of	D₄₂₁⋊#104, C₂×#27212431, #41003837, …
maximal quotient of	#41003688, Q₈⋊#16809897, #41003692, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13472, 211);