Dic₂₇₂₇ #72212948

order	10908
automorphism	Aut(G) order 9817200
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₇₂₇

Splittings

derived, Hall	C₂₇×C₁₀₁ ⋊ C₄
Sylow	C₁₀₁ ⋊ Dic₂₇
Sylow	C₂₇ ⋊ Dic₁₀₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₇₂₇
commutator, hypocenter	C₂₇×C₁₀₁	C₄
Frattini	C₂×C₉	D₃₀₃
Fitting, cosocle	C₂×C₂₇×C₁₀₁	C₂
socle, prime radical	C₂×C₃×C₁₀₁	D₉
LCS	G ▷ C₂₇×C₁₀₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 10908

A minimal faithful permutation representation — 2 generators on 132 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);(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)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,124)(42,123)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)

Polycyclic presentation:

2^2,3,101,3^2|a^2=b,c^3=e,e^3=f,c^a=c^2e^2f^2,d^a=d^100,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

2730 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₄₅₄, Dic₈₁₈₁, Dic₁₃₆₃₅
maximal normal subgroup of	Dic₅₄₅₄
maximal quotient of	Dic₈₁₈₁, Dic₁₃₆₃₅

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");;
gap> coll := SingleCollector(F, [2, 2, 3, 101, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4^100);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2727 #72212948