Dic₁₁₃₆₂ #72275666

order	45448
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₃₆₂

Splittings

Hall	C₁₃×C₁₉×C₂₃ ⋊ Q₈
Hall	C₁₉×C₂₃ ⋊ Dic₂₆
Hall	C₁₃×C₂₃ ⋊ Dic₃₈
Hall	C₁₃×C₁₉ ⋊ Dic₄₆
Sylow	C₂₃ ⋊ Dic₄₉₄
Sylow	C₁₉ ⋊ Dic₅₉₈
Sylow	C₁₃ ⋊ Dic₈₇₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₆₈₁
commutator, socle, prime radical, cosocle	C₂×C₁₃×C₁₉×C₂₃	C₂²
Fitting	C₄×C₁₃×C₁₉×C₂₃	C₂
hypercenter	C₄	D₅₆₈₁
hypocenter	C₁₃×C₁₉×C₂₃	Q₈
LCS	G ▷ C₂×C₁₃×C₁₉×C₂₃ ▷ C₁₃×C₁₉×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 45448

A minimal faithful permutation representation — 2 generators on 63 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);(1,5,3,7)(2,6,4,8)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2^3,13,19,23|a^2=c,b^2=c,b^a=bc,d^a=d^12,e^a=e^18,f^a=f^22

Conjugacy classes & character table

11365 conjugacy classes. The character table is too large to display.

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, 13, 19, 23]);;
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.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic11362 #72275666