Dic₅₇₇₂ #72224174

order	23088
automorphism	Aut(G) order 39896064
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₇₇₂

Splittings

Hall	C₃×C₁₃×C₃₇ ⋊ Q₁₆
Hall	C₁₃×C₃₇ ⋊ Dic₁₂
Hall	C₃×C₃₇ ⋊ Dic₅₂
Hall	C₃×C₁₃ ⋊ Dic₁₄₈
Sylow	C₃₇ ⋊ Dic₁₅₆
Sylow	C₁₃ ⋊ Dic₄₄₄
Sylow	C₃ ⋊ Dic₁₉₂₄

Subgroups

	subgroup	G/H
center	C₂	D₅₇₇₂
commutator, cosocle	C₃×C₄×C₁₃×C₃₇	C₂²
Frattini	C₄	C₂×D₁₄₄₃
Fitting	C₃×C₈×C₁₃×C₃₇	C₂
socle, prime radical	C₂×C₃×C₁₃×C₃₇	D₄
hypercenter	C₈	D₁₄₄₃
hypocenter	C₃×C₁₃×C₃₇	Q₁₆
LCS	G ▷ C₃×C₄×C₁₃×C₃₇ ▷ C₂×C₃×C₁₃×C₃₇ ▷ C₃×C₁₃×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 23088

A minimal faithful permutation representation — 2 generators on 69 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2^4,3,13,37|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^12,g^a=g^36

Conjugacy classes & character table

5775 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₅₄₄
maximal normal subgroup of	Dic₁₁₅₄₄

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, 2, 2, 3, 13, 37]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5772 #72224174