#72234409

order	16384
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₃₂ · Schur multiplier (H₂) = C₁₆²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₃₂	C₁₆²
commutator	C₁₆	C₂×C₁₆×C₃₂
Frattini, cosocle	#72234241	C₂³
socle	C₂²	#72234269
prime radical	C₂³	#72210694
LCS	G ▷ C₁₆ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ #72234241 ▷ C₄²×C₁₆ ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2159 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:

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

Polycyclic presentation:

2^14|a^2=d,b^2=e,c^2=fjl,d^2=g,e^2=h,f^2=iln,g^2=j,h^2=k,i^2=l,j^2=m,k^2=n,l^2=n,m^2=n,b^a=bf,c^b=cl,d^b=di,e^a=eiln,e^c=en,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.10*F.12);;
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, 11, F.14);;
gap> SetPower(coll, 12, F.14);;
gap> SetPower(coll, 13, F.14);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.12);;
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, 3, F.5*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);