H(13,𝔽₂) #72274298

order	8192
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂¹² · Schur multiplier (H₂) = C₂⁶⁵

Aliases

extraspecial	2₊¹⁺¹²
Heisenberg	H(13,𝔽₂)

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂	C₂¹²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ G
LECS	G ▷ C₂ ▷ C₁

488205468839 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 12 generators on 128 points:

(2,9)(10,30)(11,31)(12,32)(13,33)(14,34)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(110,121)(111,122)(112,123)(113,124)(114,125)(126,128);(1,2)(3,10)(4,11)(5,12)(6,13)(7,14)(8,9)(15,30)(16,35)(17,36)(18,37)(19,38)(20,31)(21,39)(22,40)(23,41)(24,32)(25,42)(26,43)(27,33)(28,44)(29,34)(45,65)(46,66)(47,67)(48,68)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,69)(56,70)(57,71)(58,81)(59,82)(60,83)(61,72)(62,73)(63,84)(64,74)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,110)(92,111)(93,112)(94,113)(95,106)(96,107)(97,108)(98,114)(99,109)(115,121)(116,122)(117,123)(118,124)(119,126)(120,125)(127,128);(3,15)(10,30)(16,45)(17,46)(18,47)(19,48)(35,65)(36,66)(37,67)(38,68)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(91,115)(92,116)(93,117)(94,118)(110,121)(111,122)(112,123)(113,124)(119,127)(126,128);(1,3)(2,10)(4,16)(5,17)(6,18)(7,19)(8,15)(9,30)(11,35)(12,36)(13,37)(14,38)(20,45)(21,49)(22,50)(23,51)(24,46)(25,52)(26,53)(27,47)(28,54)(29,48)(31,65)(32,66)(33,67)(34,68)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(55,85)(56,86)(57,87)(58,91)(59,92)(60,93)(61,88)(62,89)(63,94)(64,90)(69,100)(70,101)(71,102)(72,103)(73,104)(74,105)(81,110)(82,111)(83,112)(84,113)(95,115)(96,116)(97,117)(98,119)(99,118)(106,121)(107,122)(108,123)(109,124)(114,126)(120,127)(125,128);(4,20)(11,31)(16,45)(21,55)(22,56)(23,57)(35,65)(39,69)(40,70)(41,71)(49,85)(50,86)(51,87)(58,95)(59,96)(60,97)(75,100)(76,101)(77,102)(81,106)(82,107)(83,108)(91,115)(92,116)(93,117)(98,120)(110,121)(111,122)(112,123)(114,125)(119,127)(126,128);(1,4)(2,11)(3,16)(5,21)(6,22)(7,23)(8,20)(9,31)(10,35)(12,39)(13,40)(14,41)(15,45)(17,49)(18,50)(19,51)(24,55)(25,58)(26,59)(27,56)(28,60)(29,57)(30,65)(32,69)(33,70)(34,71)(36,75)(37,76)(38,77)(42,81)(43,82)(44,83)(46,85)(47,86)(48,87)(52,91)(53,92)(54,93)(61,95)(62,96)(63,98)(64,97)(66,100)(67,101)(68,102)(72,106)(73,107)(74,108)(78,110)(79,111)(80,112)(84,114)(88,115)(89,116)(90,117)(94,119)(99,120)(103,121)(104,122)(105,123)(109,125)(113,126)(118,127)(124,128);(5,24)(12,32)(17,46)(21,55)(25,61)(26,62)(36,66)(39,69)(42,72)(43,73)(49,85)(52,88)(53,89)(58,95)(59,96)(63,99)(75,100)(78,103)(79,104)(81,106)(82,107)(84,109)(91,115)(92,116)(94,118)(98,120)(110,121)(111,122)(113,124)(114,125)(119,127)(126,128);(1,5)(2,12)(3,17)(4,21)(6,25)(7,26)(8,24)(9,32)(10,36)(11,39)(13,42)(14,43)(15,46)(16,49)(18,52)(19,53)(20,55)(22,58)(23,59)(27,61)(28,63)(29,62)(30,66)(31,69)(33,72)(34,73)(35,75)(37,78)(38,79)(40,81)(41,82)(44,84)(45,85)(47,88)(48,89)(50,91)(51,92)(54,94)(56,95)(57,96)(60,98)(64,99)(65,100)(67,103)(68,104)(70,106)(71,107)(74,109)(76,110)(77,111)(80,113)(83,114)(86,115)(87,116)(90,118)(93,119)(97,120)(101,121)(102,122)(105,124)(108,125)(112,126)(117,127)(123,128);(6,27)(13,33)(18,47)(22,56)(25,61)(28,64)(37,67)(40,70)(42,72)(44,74)(50,86)(52,88)(54,90)(58,95)(60,97)(63,99)(76,101)(78,103)(80,105)(81,106)(83,108)(84,109)(91,115)(93,117)(94,118)(98,120)(110,121)(112,123)(113,124)(114,125)(119,127)(126,128);(1,6)(2,13)(3,18)(4,22)(5,25)(7,28)(8,27)(9,33)(10,37)(11,40)(12,42)(14,44)(15,47)(16,50)(17,52)(19,54)(20,56)(21,58)(23,60)(24,61)(26,63)(29,64)(30,67)(31,70)(32,72)(34,74)(35,76)(36,78)(38,80)(39,81)(41,83)(43,84)(45,86)(46,88)(48,90)(49,91)(51,93)(53,94)(55,95)(57,97)(59,98)(62,99)(65,101)(66,103)(68,105)(69,106)(71,108)(73,109)(75,110)(77,112)(79,113)(82,114)(85,115)(87,117)(89,118)(92,119)(96,120)(100,121)(102,123)(104,124)(107,125)(111,126)(116,127)(122,128);(7,29)(14,34)(19,48)(23,57)(26,62)(28,64)(38,68)(41,71)(43,73)(44,74)(51,87)(53,89)(54,90)(59,96)(60,97)(63,99)(77,102)(79,104)(80,105)(82,107)(83,108)(84,109)(92,116)(93,117)(94,118)(98,120)(111,122)(112,123)(113,124)(114,125)(119,127)(126,128);(1,7)(2,14)(3,19)(4,23)(5,26)(6,28)(8,29)(9,34)(10,38)(11,41)(12,43)(13,44)(15,48)(16,51)(17,53)(18,54)(20,57)(21,59)(22,60)(24,62)(25,63)(27,64)(30,68)(31,71)(32,73)(33,74)(35,77)(36,79)(37,80)(39,82)(40,83)(42,84)(45,87)(46,89)(47,90)(49,92)(50,93)(52,94)(55,96)(56,97)(58,98)(61,99)(65,102)(66,104)(67,105)(69,107)(70,108)(72,109)(75,111)(76,112)(78,113)(81,114)(85,116)(86,117)(88,118)(91,119)(95,120)(100,122)(101,123)(103,124)(106,125)(110,126)(115,127)(121,128)

Polycyclic presentation:

2^13|b^a=bm,d^c=dm,f^e=fm,h^g=hm,j^i=jm,l^k=lm

Conjugacy classes & character table

4097 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×H(13,𝔽₂), #72290098
maximal normal subgroup of	C₂×H(13,𝔽₂), #72290098
maximal quotient of	C₂×H(13,𝔽₂)

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2*F.13);;
gap> SetConjugate(coll, 4, 3, F.4*F.13);;
gap> SetConjugate(coll, 6, 5, F.6*F.13);;
gap> SetConjugate(coll, 8, 7, F.8*F.13);;
gap> SetConjugate(coll, 10, 9, F.10*F.13);;
gap> SetConjugate(coll, 12, 11, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

H(13,𝔽2) #72274298