Dic₁₁₀₇ #72211751

order	4428
automorphism	Aut(G) order 1594080
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₀₇

Splittings

derived, Hall	C₂₇×C₄₁ ⋊ C₄
Sylow	C₄₁ ⋊ Dic₂₇
Sylow	C₂₇ ⋊ Dic₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₁₀₇
commutator, hypocenter	C₂₇×C₄₁	C₄
Frattini	C₂×C₉	D₁₂₃
Fitting, cosocle	C₂×C₂₇×C₄₁	C₂
socle, prime radical	C₂×C₃×C₄₁	D₉
LCS	G ▷ C₂₇×C₄₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 4428

A minimal faithful permutation representation — 2 generators on 72 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);(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)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2^2,3,41,3^2|a^2=b,c^3=e,e^3=f,c^a=c^2e^2f^2,d^a=d^40,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

1110 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₂₁₄, Dic₃₃₂₁, Dic₅₅₃₅, …
maximal normal subgroup of	Dic₂₂₁₄
maximal quotient of	Dic₃₃₂₁, Dic₅₅₃₅, Dic₇₇₄₉, …

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, 41, 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.4^40);;
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);

Dic1107 #72211751