Dic₄₃₄₇ #72215792

order	17388
automorphism	Aut(G) order 20656944
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₄₇

Splittings

derived, Hall	C₇×C₂₃×C₂₇ ⋊ C₄
Hall	C₂₃×C₂₇ ⋊ Dic₇
Hall	C₇×C₂₇ ⋊ Dic₂₃
Hall	C₇×C₂₃ ⋊ Dic₂₇
Sylow	C₂₇ ⋊ Dic₁₆₁
Sylow	C₂₃ ⋊ Dic₁₈₉
Sylow	C₇ ⋊ Dic₆₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₃₄₇
commutator, hypocenter	C₇×C₂₃×C₂₇	C₄
Frattini	C₂×C₉	D₄₈₃
Fitting, cosocle	C₂×C₇×C₂₃×C₂₇	C₂
socle, prime radical	C₂×C₃×C₇×C₂₃	D₉
LCS	G ▷ C₇×C₂₃×C₂₇
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 17388

A minimal faithful permutation representation — 2 generators on 61 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);(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)(33,38)(34,37)(35,36)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2^2,3,7,23,3^2|a^2=b,c^3=f,f^3=g,c^a=c^2f^2g^2,d^a=d^6,e^a=e^22,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

4350 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₆₉₄, Dic₁₃₀₄₁
maximal normal subgroup of	Dic₈₆₉₄
maximal quotient of	Dic₁₃₀₄₁

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, 7, 23, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4347 #72215792