#72234360

order	16384
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₃₂ · Schur multiplier (H₂) = C₁₆×C₃₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₃₂	C₁₆²
commutator	C₁₆	C₂×C₁₆×C₃₂
Frattini, cosocle	#72234231	C₂³
socle	C₂²	#72234262
prime radical	C₂³	#72210664
LCS	G ▷ C₁₆ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ #72234231 ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

2191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 544
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 3 generators on 544 points:

(1,2,3,7,4,8,11,17,5,9,12,18,14,20,23,27,6,10,13,19,15,21,24,28,16,22,25,29,26,30,31,32)(33,34,37,45)(35,51,53,120,56,123,137,288,58,126,138,289,59,124,139,290)(36,44,61,94)(38,73,74,194,76,196,186,262)(39,47,69,102)(40,49,71,104)(41,48,72,106)(42,50,70,105)(43,85,87,207,89,209,220,302,92,210,221,303,90,212,222,127)(46,107,108,202,110,271,263,77)(52,119,128,279,131,282,306,461,133,285,307,462,134,283,308,463)(54,140,141,329,143,293,135,323,145,330,324,378,151,339,325,498)(55,122,136,257,147,231,326,258,148,230,327,259,149,91,211,100)(57,125,67,177,146,295,179,379,152,299,180,380,153,298,181,381)(60,93,155,233)(62,165,166,366,168,368,358,438)(63,96,161,239)(64,98,163,241)(65,97,164,243)(66,99,162,242)(68,182,150,276,184,384,78,112)(75,195,198,392,200,393,386,427)(79,113,187,264)(80,114,189,267)(81,115,188,266)(82,116,192,269)(83,118,190,268)(84,117,191,103)(86,206,213,395,215,397,407,478,218,398,408,479,216,400,409,300)(88,223,224,334,225,405,219,338,227,421,419,142,229,331,420,333)(95,244,245,374,247,447,439,169)(101,199,228,328,261,203,111,185)(109,270,273,347,274,458,457,156)(121,197,291,335,292,178,287,336,294,382,474,332,297,383,475,144)(129,309,310,487,312,466,304,481,314,488,482,504,320,497,483,544)(130,281,305,433,316,418,484,434,317,417,485,435,318,217,399,237)(132,284,159,349,315,468,351,505,321,472,352,506,322,471,353,507)(154,232,340,423)(157,235,346,430)(158,236,345,429)(160,354,319,452,356,510,170,249)(167,367,370,518,372,519,512,530)(171,250,359,440)(172,251,361,443)(173,252,360,442)(174,253,364,445)(175,255,362,444)(176,254,363,240)(183,286,337,431,385,473,201,234)(193,389,390,480,226,208,272,402,391,403,256,401,422,404,455,301)(204,277,387,428)(205,278,388,265)(214,410,411,492,412,525,406,496,414,528,526,311,416,489,527,491)(238,371,415,486,437,375,248,357)(246,446,449,502,450,538,537,341)(260,394,296,476,456,348,275,344)(280,369,464,493,465,350,460,494,467,508,540,490,470,509,541,313)(342,425,501,533)(343,426,500,532)(355,459,495,534,511,539,373,424)(365,515,516,543,413,396,448,522,517,523,432,521,529,524,535,477)(376,453,513,531)(377,454,514,441)(436,520,469,542,536,503,451,499);(33,35,38,54)(34,43,46,88,48,90,111,225,50,92,110,227,117,89,228,229)(36,52,62,129)(37,67,68,178,70,180,184,383)(39,55,75,142)(40,57,77,144)(41,59,78,143)(42,58,76,145)(44,86,95,214,97,216,248,412,99,218,247,414,254,215,415,416)(45,100,101,256,103,257,108,272,105,258,261,390,106,259,263,455)(47,51,109,121,114,124,275,292,115,126,274,294,278,123,296,297)(49,91,112,226,116,230,196,391,118,231,276,422,113,122,73,193)(53,107,135,192,138,271,325,187)(56,150,151,84)(60,127,156,301)(61,159,160,350,162,352,356,509)(63,130,167,311)(64,132,169,313)(65,134,170,312)(66,133,168,314)(69,85,183,208,188,210,385,404)(71,139,185,324,190,137,328,141)(72,181,186,382,191,179,74,197)(79,146,199,332)(80,149,201,333)(81,148,200,334)(82,153,203,335)(83,152,202,336)(87,182,219,268,220,194,224,269,221,384,420,104,222,262,419,264)(93,125,234,293,235,298,393,330,236,299,431,339,277,295,195,140)(94,237,238,432,240,433,245,448,242,434,437,516,243,435,439,535)(96,119,246,280,251,283,451,465,252,285,450,467,454,282,469,470)(98,217,249,413,253,417,368,517,255,418,452,529,250,281,165,365)(102,177,260,378,265,379,273,323,266,380,456,329,267,381,457,498)(120,286,287,429,288,392,291,430,289,473,475,233,290,427,474,428)(128,244,304,364,307,447,483,359)(131,319,320,176)(136,270,223,346,327,458,421,387)(147,337,338,205)(154,300,341,477)(155,211,344,405,345,326,476,331)(157,303,348,480)(158,302,347,402)(161,206,355,396,360,398,511,524)(163,308,357,482,362,306,486,310)(164,353,358,508,363,351,166,369)(171,315,371,490)(172,318,373,491)(173,317,372,492)(174,322,375,493)(175,321,374,494)(189,212,386,403,388,209,198,389)(204,207,394,401)(213,354,406,444,407,366,411,445,408,510,527,241,409,438,526,440)(232,284,424,466,425,471,519,488,426,472,534,497,453,468,367,309)(239,349,436,504,441,505,449,481,442,506,536,487,443,507,537,544)(279,459,460,532,461,518,464,533,462,539,541,423,463,530,540,531)(305,446,410,501,485,538,528,513)(316,495,496,377)(340,399,499,525,500,484,542,489)(342,479,503,543)(343,478,502,522)(361,400,512,523,514,397,370,515)(376,395,520,521);(33,36,60,154,40,64,80,172,41,65,157,342,82,174,81,173,42,66,158,343,83,175,205,377,84,176,204,376,79,171,39,63)(34,44,93,232,49,98,114,251,48,97,235,425,116,253,115,252,50,99,236,426,118,255,278,454,117,254,277,453,113,250,47,96)(35,52,127,300,57,132,149,318,59,134,303,479,153,322,148,317,58,133,302,478,152,321,147,316,56,131,207,395,146,315,55,130)(37,61,155,340,71,163,189,361,72,164,346,501,192,364,188,360,70,162,345,500,190,362,388,514,191,363,387,513,187,359,69,161)(38,62,156,341,77,169,201,373,78,170,348,503,203,375,200,372,76,168,347,502,202,374,337,495,150,319,394,520,199,371,75,167)(43,86,125,284,91,217,124,283,90,216,298,471,230,417,126,285,92,218,299,472,231,418,123,282,89,215,295,468,122,281,51,119)(45,94,233,423,104,241,267,443,106,243,430,533,269,445,266,442,105,242,429,532,268,444,265,441,103,240,428,531,264,440,102,239)(46,95,234,424,112,249,275,451,111,248,393,519,196,368,274,450,110,247,431,534,276,452,296,469,228,415,195,367,73,165,109,246)(53,128,85,206,67,159,211,399,139,308,212,400,181,353,327,485,138,307,210,398,180,352,326,484,137,306,209,397,179,351,136,305)(54,129,301,477,144,313,333,491,143,312,480,543,335,493,334,492,145,314,402,522,336,494,338,496,151,320,401,521,332,490,142,311)(68,160,344,499,185,357,386,512,186,358,458,538,271,447,385,511,184,356,476,542,328,486,198,370,74,166,270,446,107,244,183,355)(87,213,177,349,100,237,290,463,222,409,381,507,259,435,289,462,221,408,380,506,258,434,288,461,220,407,379,505,257,433,120,279)(88,214,293,466,226,413,292,465,225,412,330,488,391,517,294,467,227,414,339,497,422,529,297,470,229,416,140,309,193,365,121,280)(101,238,427,530,262,438,457,537,263,439,473,539,384,510,456,536,261,437,392,518,194,366,273,449,108,245,286,459,182,354,260,436)(135,304,208,396,178,350,405,525,324,482,403,523,382,508,421,528,325,483,404,524,383,509,331,489,141,310,389,515,197,369,223,410)(219,406,378,504,256,432,474,540,419,526,498,544,455,535,475,541,420,527,329,487,390,516,291,464,224,411,323,481,272,448,287,460)

Polycyclic presentation:

2^14|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=iln,g^2=j,h^2=k,i^2=l,j^2=m,l^2=n,b^a=bf,d^b=di,e^a=eiln,e^d=eln,g^b=gl,g^e=gn,h^a=hln,h^d=hn,j^b=jn,k^a=kn

Conjugacy classes & character table

1504 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	H(3,ℤ/32)
maximal normal subgroup of	H(3,ℤ/32)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m", "n");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.12*F.14);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.12);;
gap> SetPower(coll, 10, F.13);;
gap> SetPower(coll, 12, F.14);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.12*F.14);;
gap> SetConjugate(coll, 5, 4, F.5*F.12*F.14);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.14);;
gap> SetConjugate(coll, 8, 1, F.8*F.12*F.14);;
gap> SetConjugate(coll, 8, 4, F.8*F.14);;
gap> SetConjugate(coll, 10, 2, F.10*F.14);;
gap> SetConjugate(coll, 11, 1, F.11*F.14);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);