Dic₃₂₆₇ #72213673

order	13068
automorphism	Aut(G) order 12937320
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₂₆₇

Splittings

derived, Hall	C₂₇×C₁₂₁ ⋊ C₄
Sylow	C₁₂₁ ⋊ Dic₂₇
Sylow	C₂₇ ⋊ Dic₁₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₂₆₇
commutator, hypocenter	C₂₇×C₁₂₁	C₄
Frattini	C₂×C₉×C₁₁	D₃₃
Fitting, cosocle	C₂×C₂₇×C₁₂₁	C₂
socle, prime radical	C₂×C₃×C₁₁	D₉₉
LCS	G ▷ C₂₇×C₁₂₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 152
transitive degree	μ_t(G) = 13068

A minimal faithful permutation representation — 2 generators on 152 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);(1,3,2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,152)(34,151)(35,150)(36,149)(37,148)(38,147)(39,146)(40,145)(41,144)(42,143)(43,142)(44,141)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,132)(54,131)(55,130)(56,129)(57,128)(58,127)(59,126)(60,125)(61,124)(62,123)(63,122)(64,121)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)

Polycyclic presentation:

2^2,3,11,3,11,3|a^2=b,c^3=e,d^11=f,e^3=g,c^a=c^2e^2g^2,d^a=d^10f^10,e^a=e^2g^2,f^a=f^10,g^a=g^2

Conjugacy classes & character table

3270 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₅₃₄, Dic₉₈₀₁, Dic₁₆₃₃₅
maximal normal subgroup of	Dic₆₅₃₄
maximal quotient of	Dic₉₈₀₁, Dic₁₆₃₃₅

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

Dic3267 #72213673