Dic₂₂₂₆ #72212574

order	8904
automorphism	Aut(G) order 5556096
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₂₂₆

Splittings

Hall	C₃×C₇×C₅₃ ⋊ Q₈
Hall	C₇×C₅₃ ⋊ Dic₆
Hall	C₃×C₅₃ ⋊ Dic₁₄
Sylow	C₅₃ ⋊ Dic₄₂
Hall	C₃×C₇ ⋊ Dic₁₀₆
Sylow	C₇ ⋊ Dic₃₁₈
Sylow	C₃ ⋊ Dic₇₄₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₁₁₃
commutator, socle, prime radical, cosocle	C₂×C₃×C₇×C₅₃	C₂²
Fitting	C₃×C₄×C₇×C₅₃	C₂
hypercenter	C₄	D₁₁₁₃
hypocenter	C₃×C₇×C₅₃	Q₈
LCS	G ▷ C₂×C₃×C₇×C₅₃ ▷ C₃×C₇×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 8904

A minimal faithful permutation representation — 2 generators on 71 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,11)(13,18)(14,17)(15,16)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^3,3,7,53|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^52

Conjugacy classes & character table

2229 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₄₅₂, Dic₆₆₇₈, Dic₁₁₁₃₀, …
maximal normal subgroup of	Dic₄₄₅₂
maximal quotient of	Dic₆₆₇₈, Dic₁₁₁₃₀, Dic₁₅₅₈₂

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, 2, 3, 7, 53]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2226 #72212574